Copyright: Isoptikon 1.1


Isoptikon  Help Index

What do you want to know?

Why use this application

How to use this application

Tutorials

Sample documents

General on Euclidean Geometry

Theory of isoptic cubics

Commands on main menus

System menu

File menu

Edit menu

View menu

Drawing menu

Isoptics menu

Settings menu

Help menu

Commands on secondary menus (submenus)

Segments menu

Lines menu

Triangles menu

Parallelograms menu

Angles menu

Circular Arcs menu

Circles menu

Tangents-Bissectrices menu

Commands on Toolbars

Toolbar

Status Bar

Line thickness

Color tools

Font tools

Modifying Objects

Tiling the plane with objects

Status Bar

Dialogs used by the application

File Open dialog box

File Save As dialog box

Drawing Parameters dialog box

Drawing Sheet dialog box

Graph dialog box

Import picture dialog box

Export picture dialog box

Through constants dialog box

Half-component pts dialog box

Through a,b,c dialog box

Change fixed dialog-box

Point-mark-figure-selection dialog-box


Tutorials

All tutorials are documents produced by the application (of type iso).

They are contained in the folder  Tutorials, a sub-folder of the Application’s folder, named Isoptikon folder.

Presently the Tutorials-folder contains the following documents:

My_Objects A sample-document introducing the various objects which can be constructed in a document of this application.

A4_pages A sample-document indicating the formation of A4(210x297 mm)-format pages.

Segments_Lines A sample-document constructing segments and lines.

Angles_Triangles A sample-document constructing angles and triangles.

Polygons_Circles A sample-document constructing polygons and circles.

Arcs A sample-document constructing circular arcs.

Graphs A sample-document constructing graphs.

Isoptics A sample-document constructing isoptic cubics.

Pythagoras1 A sample-document constructing the drawing of Pythagora’s theorem.

Pythagoras2 A sample-document suggesting a proof of Pythagora’s theorem.

Pythagoras3 A sample-document suggesting Pappus proof of Pythagora’s theorem.

Pappus A sample-document suggesting Pappus generalization of Pythagora’s theorem.

Triangle_elements A sample-document visualizing important triangle-related-elements.

Morley A sample-document constructing Morley’s-theorem-drawing.


samples
Sample documents

All Sample-documents are documents produced by the application (of type iso).

They are contained in the folder  Samples, a sub-folder of the Application’s folder, named Isoptikon folder.

Presently the Samples-folder contains the following documents:

Elementary_facts A sample-document containing statements, drawings and proofs of some elemntary propositions, deriving directly from the axioms of Euclidean Geometry.

Triangle A sample-document constructing important elements of a triangle.

Fermat A sample-document constructing the drawing of Fermat’s theorem.

Simson A sample-document constructing a drawing of Simson’s theorem.

Euler A sample-document constructing points of the Euler circle and line.

Triangle_constructions A sample-document constructing triangles .

Tangent_circles A sample-document constructing tangent circles .

Equations_solving A sample-document solving graphically equations.

Isoptic_kinds A sample-document, where various kinds of isoptics are demonstrated.

Isoptic_configuration A sample-document, showing various isoptic-configurations.


Modifying objects

All objects defined in the application can be modified, in their sizes and shapes, by using the selection handles at their vetices (look at selection-command).

These modifications however destroy the relations between different shapes.

If you modify a segment, while holding down the ctrl-key, the direction of the segment remains the same and consequently the segment is simply extended. What changes is only its length.

The same thing happens also with the legs of an angle. By holding down the ctrl-key, you extend its legs without to modify the angle.

If Match-points is activated and you modify the objects, by dragging their corners, then releasing near some other corner of another object, causes identification of the under modification corner with the next lying corner of the other object.

For extension of a segment by its very same length (doubling) look at Parallel to _.


Tiling with objects
Tiling the plane with objects

You can fill rectangular parts of the plane with copies of any object, defined by the application.

For this, select first the object to be used as a tile.

Then, holding down the shift-key press any of the four arrow-keys. Apply repeatedly this action.

0) This is the standard tiling, works for all kinds of objects and uses their enclosing rectangles: neighbor tiles have one side of their respective enclosing rectangles in common.

For special shapes there are important variants:

1) For segments: instead of tiling, the corresponding arrows produce the segment on its respective side by its very length. Up and down arrows, create copies of the segment on top and below it. To get the standard tiling with segments, hold down both the ctrl-key and the shift-key, while pressing any of the arrow-keys.

2) For lines: instead of tiling, the up/down arrows produce parallel lines at distance k, from the selected line. k is the homothety-modulus and can be set by the appropriate field of the Drawing Parameters dialog box.

3) For angles: the left/right arrows produce copies, using the first side of the angle, so that the new angle’s first side is collinear to the first side of the selected angle. The up/down arrows do analogous operations using the second side of the selected side.

4) For angles, there is still another option, producing rotations of the selected angle x, by the very same angle x, around its vertex. Simply press the F9-key (without pressing any other key). To produce the supplementary angle: , hold down both the ctrl-key and the shift-key, while pressing the F10-key. To get the standard tiling with angles, hold down both the ctrl-key and the shift-key, while pressing any of the arrow-keys.

5) For triangles: left/right arrows produce copies of the selected triangle, along its first side. Up/down arrows produce copies along its second side. To get the standard tiling with triangles, hold down both the ctrl-key and the shift-key, while pressing any of the arrow-keys.

6) For parallelograms: left/right arrows, produce copies of the selected parallelogram, along its first side. Up/down-arrows, produce copies along its second side. To get the standard tiling with parallelograms, hold down both the ctrl-key and the shift-key, while pressing any of the arrow-keys.

7) For regular-polygons: left/right arrows, produce copies of the selected polygon, mirrored to its first side. Up/down-arrows, produce copies mirrored to its second side. The well-known tiling of the plane by equilateral triangles or regular hexagons, can be obtained this way.

To get the standard tiling with regular polygons, hold down both the ctrl-key and the shift-key, while pressing any of the arrow-keys.

8) For circles: The center A and its circumference anchor point B, define an oriented segment x=AB. The arrow-keys produce copies of the circle along this segment and its orthogonal direction. To get the standard tiling with circles, hold down both the ctrl-key and the shift-key, while pressing any of the arrow-keys.

There is still another option to tile the plane through successive Reflexions of a shape on its sides.


File Print command
Print command (File menu)

Use this command to print a document.   This command presents a Print dialog box, where you may specify the range of pages to be printed, the number of copies, the destination printer, and other printer setup options.

Shortcuts

Keys:	CTRL+P


Print dialog box

The following options allow you to specify how the document should be printed:

Printer

This is the active printer and printer connection.  Choose the Setup option to change the printer and printer connection.

Setup

Displays a Print Setup dialog box, so  you can select a printer and printer connection.

Print Range

Specify the pages you want to print:

All	Prints the entire document.
Selection	Prints the currently selected text.
Pages	Prints the range of pages you specify in the From and To boxes.
Copies

Specify the number of copies you want to print for the above page range.

Collate Copies

Prints copies in page number order, instead of separated multiple copies of each page.

Print Quality

Select the quality of the printing.  Generally, lower quality printing takes less time to produce.


Print Progress Dialog

The Printing dialog box is shown during the time that the application is sending output to the printer.  The page number indicates the progress of the printing.

To abort printing, choose Cancel.


File Print Preview command
Print Preview command (File menu)

Use this command to display the active document as it would appear when printed.  When you choose this command, the main window will be replaced with a print preview window in which one or two pages will be displayed in their printed format.  The print preview toolbar offers you options to view either one or two pages at a time; move back and forth through the document; zoom in and out of pages; and initiate a print job.


Print Preview toolbar

The print preview toolbar offers you the following options:

Print

Bring up the print dialog box, to start a print job.

Next Page

Preview the next printed page.

Prev Page

Preview the previous printed page.

One Page / Two Page

Preview one or two printed pages at a time.

Zoom In

Take a closer look at the printed page.

Zoom Out

Take a larger look at the printed page.

Close

Return from print preview to the editing window.


File Print Setup command
Print Setup command (File menu)

Use this command to select a printer and a printer connection.  This command presents a Print Setup dialog box, where you specify the printer and its connection.


Print Setup dialog box

The following options allow you to select the destination printer and its connection.

Printer

Select the printer you want to use.  Choose the Default Printer; or choose the Specific Printer option and select one of the current installed printers shown in the box.  You install printers and configure ports using the Windows Control Panel.

Orientation

Choose Portrait or Landscape.

Paper Size

Select the size of paper that the document is to be printed on.

Paper Source

Some printers offer multiple trays for different paper sources.  Specify the tray here.

Options

Displays a dialog box where you can make additional choices about printing, specific to the type of printer you have selected.

Network...

Choose this button to connect to a network location, assigning it a new drive letter.


File Page Setup Command
Page Setup command (File menu)

<< Write application-specific help here. >>


Angles  menu commands

The Angles menu offers the following commands, relateted to the construction of angles:

Start with side-1	Tool used to draw angles (in two stages).
Viewing a segment	Tool used to draw angles (in two stages), viewing the segment AB, defined in the first stage.
Fixed	Tool used to draw angles of constant measure and equal legs.
Change fixed	Displays the dialog, by means of which you can set the angle-measure of the tool of the previous item.
Divide angle	Tool to divide an angle in equal parts (default=2).


Start with side-1  command (Angles menu)

Use this command to create a new angle by defining its first side, say x=AB, and its second side, say y=BC. The angle-vertex is at B.

Operation: 	In two stages:

1st stage (draws): 	Click somewhere, say A,...drag...release, say at B.

2nd stage (draws): 	Click somewhere, say B1,...drag...release, say at C.

Shortcuts

mouse	click the angle-tool on the toolbar

Comments:

In the first stage you draw a segment. All rules of segment-drawing apply in this case.

In the second stage, you draw the second side of the angle, starting at B.

Match points enabled:

If pre-existing points nearby, readjustement of all three points A, B and C.


Viewing a segment  command (Angles menu)

Use this command to construct an angle, viewing a segment, say x=AB.

Operation: 	In two stages:

1st stage (draws): 	Click somewhere, say A,...drag...release, say at B.

2nd stage (draws): 	Click somewhere, say B1,...drag...release, say at C.

Comments:

In the first stage you draw a segment, say x=AB. All rules of segment-drawing apply in this case.

In the second stage, you define the point C, corner of the angle, viewing the segment x. The application inserts the segments CA, CB and erases x, defined in the first stage.

Match points enabled:

If pre-existing points nearby, readjustement of all three points A, B and C.


Fixed  command (Angles menu)
Fixed  (Angles menu)

Use this command to construct an angle, say x of fixed measure, say k.

Operation: 	In one stages:

1st stage (draws): 	Click somewhere, say A,...drag...release, say at B.

Comments:

Draws a positively oriented (counterclockwise) angle with equal legs. y=AB is the first side.

If you hold down the ctrl-key, the angle is negatively oriented (clockwise).

If you hold down the shift-key, the angle becomes a right angle on the segment y=AB.

The measure k of the angle, is by default 30 degrees and can be set by the user, through the Change fixed dialog.

Match points enabled:

If pre-existing points nearby, readjustement of both A and B.


Change fixed command (Angles menu)
Change fixed (Angles menu)

Displays a dialog-box, by means of which you can set the angle-measure of the tool of the previous item Fixed.

The dialog contains three radio-buttons by which you select the format in which you enter the angle:

1) 	In degrees

2) 	in radians

3) 	as a fraction of pi (f.e. pi/4)

After you make your choice using these radio-buttons, you enter the desired angle in the field, using the corresponding format. F.e. pi/10 if you selected 3).


Divide angle  command (Angles menu)

Use this command to divide an existing angle/polygon-angle, say x, in k equal parts (default k=2).

Operation: 	In one stage:

1st stage (selects): 	Click somewhere to select an angle/polygon-angle, say x.

Comments:

The application inserts k-1 lines, which divide the selected angle x in k equal parts.

You can select interior or exterior angle-division, by clicking with the mouse near the vertex of the angle inside or outside, respectively.

You can set the number k of parts of the angle through the appropriate field in the Drawing Parameters dialog, in the Settings menu.


Circlular Arcs  menu commands

The Circular Arcs menu offers the following commands, relateted to the construction of circular arcs:

Simple arc	Tool used to draw circular arcs.
Arc180	Tool used to draw circular arcs of 180 degrees.
Arc on angle	Tool used to draw circular arcs, that fit into an existing angle.
Arc H-Line	Tool used to draw hyperbolic-lines, which are half-circles sitting orthogonal to the x-axis.
Arc H-Segment	Tool used to draw hyperbolic-segments, which are arcs of circles sitting orthogonal to the x-axis.
Arc H-Next	Tool used to draw hyperbolic-lines, which are next-parallel to a given hyperbolic line.
Arc H-Orthogonal	Tool used to draw  a hyperbolic-segment, which is orthogonal to another hyperbolic segment or hyperbolic line.


Simple arc  command (Circular arc menu)
Simple arc  command (Circular arcs  menu)

Use this command to create a circular arc x=BC, lying on a circle of radius y=AB.

Operation: 	In two stages:

1st stage (draws): 	Click somewhere, say A,...drag...release, say at B.

2nd stage (draws): 	Click somewhere, say B1,...drag...release, say at C1.

Comments:

In the first stage you draw a segment, say x=AB. All rules of segment-drawing apply in this case.

A is the center of the circle, x is the radius of the circle supporting the arc. B is one end-point of the arc.

In the second stage, the radius x is erased and the second end-point C of the arc is calculated.

C is defined to be the projection of the release-point C1 on the circle x.

Match points enabled:

If pre-existing points nearby, readjustement of A, B and C1.


Arc180  command (Circular arcs  menu)

Use this command to construct 180-degrees-arcs (half-circles) on diameters, say x=AB.

Operation: 	In one stage:

1st stage (draws): 	Click somewhere, say A,...drag...release, say at B.

Comments:

The application inserts the half-circle on the left of the directed segment x=AB. It uses the point A as center and x as radius of the supporting circle. If, while dragging, you hold down the ctrl-key, then x is used as a diameter of the supporting circle. Again the half-circle is on the left of the directed segment.

Match points enabled:

If pre-existing points nearby, readjustement of both A and B.


Arc on angle  command (Circular arcs menu)

Use this command to construct a circular arc, that fits into an existing angle, say x.

Operation: 	In one stage:

1st stage (selects): 	Click somewhere, say A, to select an existing angle, say x.

Comments:

The application inserts a circular arc, say y, that fits into x, and whose supporting circle, passes through the point A.

Match points enabled:

No effect.


Arc H line  command (Circular arcs  menu)

Use this command to construct a hyperbolic line, which is a half-circle, orthogonal to the x-axis, passing through two points, say A and B.

Operation: 	In one stage:

1st stage (draws): 	Click somewhere, say A,...drag...release, say at B.

Comments:

The application inserts the half-circle passing through A, B and orthogonal to the x-axis.

If you want to fix A, B, do it before to draw the hyperbolic-line.

This is a line of the Poincare-model of hyperbolic-geometry.

Match points enabled:

If pre-existing points nearby, readjustement of both A and B.


Arc H segment  command (Circular arcs  menu)

Use this command to construct hyperbolic-segments, which are arcs of half-circles, sitting orthogonally on the x-axis.

Operation: 	In one stage:

1st stage (draws): 	Click somewhere, say A,...drag...release, say at B.

Comments:

The application inserts the hyperbolic-segment, passing through A and B.

This is a line-segment of the Poincare-model of hyperbolic-geometry.

Match points enabled:

If pre-existing points nearby, readjustement of both A and B.


Arc H next  command (Circular arcs  menu)

Use this command to construct hyperbolic-half-lines, which are next-parallels to an existing hyperbolic-line, say x.

Operation: 	In two stages:

1st stage (selects): 	Click somewhere to select a hyperbolic-line, say x.

2nd stage (draws): 	Click somewhere, say A,...drag...release, say at B.

Comments:

The application inserts the two hyperbolic-half-lines, which are next-parallel to the selected x.

These are next-parallels to a line-segment of the Poincare-model of hyperbolic-geometry.

Match points enabled:

If pre-existing points nearby, readjustement of A.


Arc H orthogonal  command (Circular arcs  menu)

Use this command to construct a hyperbolic-segment, orthogonal to another existing hyperbolic segment/line, say x,  and passing through a point, say B.

Operation: 	In two stages:

1st stage (selects): 	Click somewhere to select a hyperbolic segment/line, say x.

2nd stage (draws): 	Click somewhere, say A,...drag...release, say at B.

Comments:

The tool continues with its second stage: You can continue to draw orthogonal hyperbolic segments to the same object x, selected in the first stage.

You can turn it to its first stage (in order to select another x) by holding down the ctrl-key, while you release the mouse (defining B).

This is a orthogonal segment to a line-segment of the Poincare-model of hyperbolic-geometry.

Match points enabled:

If pre-existing points nearby, readjustement of B.


Circles  menu commands

The Circles menu offers the following commands related to the construction of circles:

General	Circle-tool, used to draw circles, centered or on their diameter.
Tangent to circle/line centered at	Tangent-centered-at, circle-tool. Constructs circles (2 or 1) tangent(s) to a given circle/line/segment and centered at a given point.
Tangent to circle/line at point	Circle-tangent-at, circle-tool.  Constructs a circle, tangent at a given circle/line/segment, at one of its points (say A) and passing through another point (say B).
Tangent to circle/line _pass 2 pts	Circle-tangent-2pts, circle-tool.  Constructs circles tangent to a circle/line/segment and passing through two other points.
Tangent_2circl/lin and pass 1pt	2-circles-tangent-1-point-passing, circle-tool.  Constructs circles tangent to two given circles/lines/segments and passing through a given point.
Tangent to _3 circles/lines	3-circles-tangent, circle-tool.  Constructs circles tangent to three given circles/lines/segments and operates as a selection-tool.
Orthogonal to circle/line_center at	Orthogonal-centered-at, circle-tool.  It constructs a circle, orthogonal to a given circle/line/segment and centered at a given point.
Orthogonal_2cir/lin pass 1pt	2-circles-orthogonal-1-pt-pass, circle-tool.  It constructs a circle, orthogonal to two given circle/line/segments and passing through a given point.
Orthogonal to cir/lin_pass 2pts	Orthogonal-2-pt-pass, circle-tool.  It constructs a circle, orthogonal to a given circle/line/segment and passing through two given points.
Radical axis_2 circles	Radical-axis, line-tool.  It constructs the radical axis of two given circles.
Inversion interchanging_2cir/lin	Inversion-interchanging, circle-tool.  It constructs the circles, defining inversions which interchange two given circles.
Apollonian circle	Apollonian-circle, circle-tool.  It constructs the Apollonian circle, of a segment AB, whose points P have a ratio |PA|/|PB|=k.
Three points passing	Three-points-passing, circle-tool.  It constructs the circle,passing through three points.


General  command (Circles menu)

Use this command to create a new circle.

Operation: 	Click somewhere, say at A,...drag...release, say at B.

Comments

A is the center. B is a point of the circumference, hence AB is the radius.

Drawing on the diameter: If you hold down the ctrl-key during definition:

A is on the circle. B is on the circle, such that AB is a diameter of the circle.

Shortcuts

mouse	click the circle-tool on the toolbar

Match points enabled:

If pre-existing points nearby, readjustement of both A and B.


Tangent to circle/line centered at  command (Circles menu)

Use this command to construct circle(s) tangent to a given cirlce/line/segment and centered at a point.

Operation: 	In two stages:

1st stage (selects): 	Click somewhere to select a circle/line/segment, say x.

2nd stage (draws): 	Click somewhere, say A,...drag...release, say at B.

x is the selected circle/line/segment.

B is the center of the circle constructed (A unimportant).

Comments:

If x is a circle: 	You get two new circles, centered at B.

If x is a line/segment:	You get one circle, centered at B.

The tool continues with its second stage: You can continue to draw circles tangent to the same object, selected in the first stage.

You can turn it to its first stage (in order to select another x) by holding down the ctrl-key, while you release the mouse (defining B).

Shortcuts

mouse	click the button on the toolbar

Match points enabled:

If pre-existing points nearby, readjustement of B.


Tangent to circle/line at point  command (Circles menu)

Use this command to construct a circle tangent to an already existing cirlce/line/segment (say x) at a point (say A) of x and passing through another point B.

Operation: 	In two stages:

1st stage (selects): 	Click somewhere to select a circle/line/segment (say x).

2nd stage (draws): 	Click somewhere (say A1)...drag...release (say at B).

x is the selected circle/line/segment.

A (calculated by the application) is the point of x, lying next to A1.

Comments:

The constructed circle (say y) passes through A, B and contacts x at A.

It is important where you first click in the second stage (point A1).

The application calculates the point of contact, A, to be the point of x, lying next to A1.

The tool continues with its second stage: You can continue to draw circles tangent to the same object, selected in the first stage.

You can turn it to its first stage (in order to select another x) by holding down the ctrl-key, while you release the mouse (defining B).

Match points enabled:

If pre-existing points on x nearby A1, readjustement of A only.

Thus, you can draw a circle tangent to x at some pre-defined point of x, say C, by clicking A1 near C.


Tangent to circle/line pass 2pts  command (Circles menu)

Use this command to construct circles tangent to an already existing cirlce/line/segment, say x and passing through two other points, say A and B.

Operation: 	In two stages:

1st stage (selects): 	Click somewhere to select a circle/line/segment, say x.

2nd stage (draws): 	Click somewhere, say A,...drag...release, say at B.

Comments:

In general (depending on the location of A, B), there are two circles constructed.

Both of them pass through A, B and are tangent to x.

The tool continues with its second stage: You can continue to draw circles tangent to the same object, selected in the first stage.

You can turn it to its first stage (in order to select another x) by holding down the ctrl-key, while you release the mouse (defining B).

Shortcuts

mouse	click the button on the toolbar

Match points enabled:

If pre-existing points nearby, readjustement of both A and B.


Tangent _2circl/lin and pass 1pt  command (Circles menu)

Use this command to construct circles tangent to two already existing cirlce/line/segments, say x and y, and passing through a point, say B.

Operation: 	In three stages:

1st stage (selects): 	Click somewhere to select a circle/line/segment, say x.

2nd stage (selects): 	Click somewhere to select a circle/line/segment, say y.

3rd stage (draws): 	Click somewhere, say A,...drag...release, say at B.

Comments:

In general (depending on the relative positions of x, y and the point A), there are four circles constructed.

All of them pass through B and are tangent to x, y.

The tool continues with its third stage: You can continue to draw circles tangent to the same objects, x, y, selected in the first two stages.

You can turn it to its first stage (in order to select other x, y) by holding down the ctrl-key, while you release the mouse (defining B).

Match points enabled:

If pre-existing points nearby, readjustement of B.


Tangent to__3 circles/lines  command (Circles menu)

Use this command to construct circles tangent to three already existing cirlce/line/segments, say x y and z..

Operation: 	In three stages:

1st stage (selects): 	Click somewhere to select a circle/line/segment, say x.

2nd stage (selects): 	Click somewhere to select a circle/line/segment, say y.

3rd stage (selects): 	Click somewhere to select a circle/line/segment, say z.

Comments:

In general (depending on the relative positions of x, y,z), there are 8 circles constructed.

All of them are tangent to the selected  x, y and z.

The tool returns to its first stage: You continue with it, by selecting three new circles.

Match points:

No-effect.


Orthogonal to circle/line center at  command (Circles menu)

Use this command to construct a circle, orthogonal to an existing cirlce/line/segment, say x, and centered at a point.

Operation: 	In two stages:

1st stage (selects): 	Click somewhere to select a circle/line/segment, say x.

2nd stage (draws): 	Click somewhere (say A)...drag...release (say at B).

x is the selected circle/line/segment.

B is the center of the circle constructed (A unimportant).

Comments:

The tool continues with its second stage: You can continue to draw circles orthogonal to the same object x, selected in the first stage.

You can turn it to its first stage (in order to select another x) by holding down the ctrl-key, while you release the mouse (defining B).

Match points enabled:

If pre-existing points nearby, readjustement of B.


Orthogonal_2cir/lin pass 1pt  command (Circles menu)

Use this command to construct a circle, orthogonal to two existing cirlce/line/segments, say x and y, and passing through a point, say A.

Operation: 	In three stages:

1st stage (selects): 	Click somewhere to select a circle/line/segment, say x.

2nd stage (selects): 	Click somewhere to select a circle/line/segment, say y.

3rd stage (draws): 	Click somewhere,say A,...drag...release, say at B.

x, y are the selected circle/line/segments.

B is on the circle, which is constructed to be orthogonal to both x, y.

Comments:

The tool continues with its third stage: You can continue to draw circles orthogonal to the same objects, x, y, selected in the first two stages.

You can turn it to its first stage (in order to select other x, y) by holding down the ctrl-key, while you release the mouse (defining B).

Match points enabled:

If pre-existing points nearby, readjustement of B.


Orthogonal to cir/lin pass 2pts command (Circles menu)
Orthogonal to cir/lin pass 2pts (Circles menu)

Use this command to construct a circle, orthogonal to an existing cirlce/line/segment, say x, and passing through two points, say A and B.

Operation: 	In two stages:

1st stage (selects): 	Click somewhere to select a circle/line/segment, say x.

2nd stage (draws): 	Click somewhere, say A,...drag...release, say at B.

x is the selected circle/line/segment.

A and B are points on the constructed circle.

Comments:

The tool continues with its second stage: You can continue to draw circles orthogonal to the same object x, selected in the first stage.

You can turn it to its first stage (in order to select another x) by holding down the ctrl-key, while you release the mouse (defining B).

Match points enabled:

If pre-existing points nearby, readjustement of both A and B.


Radical axis_2 circles command (Circles menu)
Radical axis_2 circles  (Circles menu)

Use this command to construct a line: The radical axis of two existing circles, say x and y.

Operation: 	In two stages:

1st stage (selects): 	Click somewhere to select a circle, say x.

2nd stage (selects): 	Click somewhere to select a circle, say y.

The application inserts the radical axis of x and y.

Comments:

The tool returns to its first stage: You continue with it, by selecting two new circles.

Match points enabled:

No-effect.


Inversion interchanging_2cir/lin command (Circles menu)
Inversion interchanging_2cir/lin  (Circles menu)

Use this command to construct a(two) circle(s), say a, which defines an inversion, interchanging two existing circle/line/segments, say x and y. One of x, y must be circle.

Operation: 	In two stages:

1st stage (selects): 	Click somewhere to select a circle/line/segment, say x.

2nd stage (selects): 	Click somewhere to select a circle/line/segment, say y.

Comments:

In general, there are two circles constructed. One defines an inversion, the other defines an anti-inversion, interchanging x, y, i.e. transforming x to y and vice versa.

The tool returns to its first stage: You continue with it, by selecting two new circles.

Match points enabled:

No-effect.


Apollonian circle command (Circles menu)
Apollonian circle  (Circles menu)

Use this command to construct the Apollonian circle, of an existing segment, say x.

Operation: 	In one stage:

1st stage (selects): 	Click somewhere to select a segment, say x.

The application constructs the Apollonian circle, of a segment x=AB, whose points P have a ratio |PA|/|PB|=k.

Comments:

The value of k can be set by the drawing-parameters-dialog, in the settings-menu.

The tool returns to its first stage: You continue with it, by selecting a new segment.

The ordering of A, B plays a role.

If you define the same segment with the roles of A, B interchanged, then, the new Apollonian circle is the reflexion of the former one on the middle-orthogonal line of the segment AB.

Match points enabled:

No-effect.


Three points passing command (Circles menu)
Three points passing  (Circles menu)

Use this command to construct a circle, passing through three points, say A, B and C.

Operation: 	In two stages:

1st stage (draws): 	Click somewhere (say A)...drag...release (say at B).

2nd stage (draws): 	Click somewhere (say B1)...drag...release (say at C).

In the first stage an intermediate circle is constructed, with diameter AB.

In the second stage, the circle of the first stage is modified so as to pass through all three points A, B and C (B1 is unimportant).

Comments:

The tool returns to its first stage.

Match points enabled:

If pre-existing points nearby, readjustement of all three A, B and C.


Drawing menu commands

The Drawing menu contains all the tools needed for drawings of Euclidean Geometry.

Most of these tools are grouped in submenus of related items f.e. Segments menu

Here are described only the items accessible directly from this menu.

The other commands are discussed in their corresponding submenus.

Select	Get the selection-tool, in order to pick, move, modify objects.
Polygons	Get the polygon-tool, in order to draw regular n-gons.
Set points	Get the point-tool, in order to set points unrestricted, restricted on shape contours or at intersections of contours of shapes.
Graph	Displays a dialog, where you define the formula of a function and draws the corresponding graph.
Text	Get the text-tool, in order to draw a text-box and type text there.
Restricted pict (bmp)	Get the bitmap-creation-tool in order to define portions of the drawing sheet as bitmap-images.
Restricted pict (emf)	Get the enhanced-metafile-creation-tool in order to define portions of the drawing sheet as enhanced-metafile-images.
Import picture	Displays a dialog, in order to import a picture in bitmap or enhanced metafile format, as an object of your document.
Export picture	Displays a dialog, in order to export in a single file a previously selected bmp-picture or emf-picture of your document.

More tools are contained in the (sub)menus:

Segments menu

Lines menu

Triangles menu

Parallelograms menu

Angles menu

Circular Arcs menu

Circles menu

And the advanced (for isoptic cubics drawings):

Isoptics menu

Tangents-Bissectrices menu


Drawing: selection tool
Select  command  (Drawing menu)

Use this command to get the selection tool.

Operation: 	In one stage:

1st stage (selects):	Click somewhere, say A,...drag...release, say at B.

Shortcuts

mouse	click the selection-tool on the toolbar

Keys:	CTRL+0

Comments:

A is the click-point. B is the release-point.

By clicking near the contour of a shape you can select it (snap, pick or grab).

Dragging without releasing the mouse-button, moves (translates) the object.

Clicking on the selection-marks (or handles: small black squares on the vertices) of a shape, you pick a corner and modify it.

Selecting a non-selected object, deselects any other selected objects, except if you hold down the shift-key, in which case the new object is added to the collection of the selected objects.

Clicking on a selected object, preserves the selection property of all other objects and subsequent movement is applied to all selected objects.


Drawing: Polygons tool
Polygons  command  (Drawing menu)

Use this command to draw regular polygons, centered (default) or on one-side-based (ctrl-key).

Operation: 	In one stage:

1st stage (draws):	Click somewhere, say A,...drag...release, say at B.

Shortcuts

mouse	click the polygon-tool on the toolbar

Comments:

The application inserts a regular polygon with k sides.

A is the center of the polygon. B is one vertex of the polygon.

These two points, together with the number k of sides of the polygon, completely define its shape.

If, while dragging, you hold down the ctrl-key, then the polygon is drawn, so that x=AB becomes a side of it. The polygon is constructed on the left side of the oriented segment AB.

You can change the value of k, to draw 3-,4-,...,9-gons, by pressing the corresponding number-key on the keyboard.

You can set any other side-number (3<k<61) through the appropriate field of the Drawing Parameters dialog box, in the Settings menu.

Match points enabled:

If pre-existing points nearby, readjustement of both A and B.


Drawing: Set points tool
Set points  command  (Drawing menu)

Use this command to set point-marks, anywhere or on shapes or at intersections of shapes.

The shape of point-marks can be set by displaying the Point-mark-figure-selection dialog-box.

Operation: 	In one stage:

1st stage (draws):	Click somewhere, say A.

Shortcuts

Keys:	CTRL+E

Comments:

The application inserts a point at A, marked (by default) with

This point-mark has the properties of a circle. In most cases you will use such marks for conveniance and eliminate at the end of the procedure.

To eliminate all of them at once, use the Delete All Point-marks command.

If you hold down the ctrl-key, then the next to A intersection point of two existing shapes is inserted.

If you hold down the shift-key, then the next to A point on a shape inserted.

The two last options make this tool important for precise drawing, since you are able to define points, lying exactly on shapes or at their intersections.

To locate exactly (using this tool) the actual rotation-center and the actual homothety center look at the Drawing Parameters... dialog

Match points enabled:

No effect.


Drawing: Graph  tool
Graph  command  (Drawing menu)

Use this command to display the Graph dialog box, type the formula of a function and get its graph.

Shortcuts

Keys:	CTRL+G

Comments:

The dialog-box contains 5 fields:

Set the function ...	Type there the formula defining the function.

The next two fields set respectively the left and right end of the interval, where you want the graph of the function:

Set interval’s left end:	Type interval’s left end (default: -5.0).

Set interval’s right end:	Type interval’s right end (default:  5.0).

Set number of points on graphs:	Type the number of points each graph should contain (default = 50).

Set then component-separation-tolerance:	Type a positive real number (default =7.2).

The last field is important when the function is discontinuous or very rapidly growing and consequently for small changes of x you get big differences in the values of y=f(x). When the differences of y (for two successive x’s) become larger than the constant T (named component-separation-tolerance, default = 7.2), then the application stops connecting the points and the graph appears to be disconnected.

If the gaps between the components are too big, set a bigger number of graph-points and/or set a bigger number for T.

When typing the formula of the function pay attention to the following rules:

The independent variable is 	x

Multiplication is denoted by 	*

Powers of x are denoted by 	x*x*...*x  (n times for the n-th power of x).

You can rise an expression (A1) to the power (A2)	(A1)^(A2)

but you must know that (A1) is positive for all values of x.

F.e. 	(1.0+x*x)^(1.+x) 	is a valid expression.

The absolute-value-function is denoted by 	abs(...)

The arcus-cos function is denoted by 	acos(...)

The arcus-sin function is denoted by 	asin(...)

The arcus-tan function is denoted by 	atan(...)

The cosinus function is denoted by 	cos(...)

The cosinus-hyperbolicus is denoted by 	cosh(...)

The natural logarithm is denoted by 	log(...)

The exponential function is denoted by 	exp(...)

The sinus function is denoted by 	sin(...)

The sinus-hyperbolicus is denoted by 	sinh(...)

The square-root function is denoted by 	sqrt(...)

The tangens function is denoted by 	tan(...)

The tangens hyperbolicus is denoted by 	tanh(...)


Drawing: Text  tool
Text  command  (Drawing menu)

Use this command to define a rectangle and write there some text.

Operation: 	In two stages:

1st stage (draws):	Click somewhere, say A,...drag...release, say at B.

2nd stage (enter text):	Box is active: Type your text.

If the click (point A) occurs inside an existing text-box T, then no new text box is defined. Instead, T is activated and you can modify its text.

Comments:

In the first stage you draw a rectangular box, where text will be displayed.

In the second stage you enter the text.

The text-characteristics, font, size, alignement are controled through the Font toolbar .

The color of the text is controled through the Color toolbar.

Shortcuts

Mouse:	click the button on the toolbar

Keys:	CTRL+T

Match points enabled:

If pre-existing points nearby, readjustement of both A and B.


Drawing: Restricted pict (bmp)  tool
Restricted pict (bmp)  command  (Drawing menu)

Use this command to define a rectangular portion of the drawing sheet, and build a bitmap-image of the contents of this rectangle.

Operation: 	In one stage:

1st stage (selects):	Click somewhere, say A,...drag...release, say at B.

Comments:

The rectangular box defines the enclosing box of the bitmap-image.

You can cut-paste this image to the same or any other application that accepts bitmap-images.

Match points enabled:

No effect.


Drawing: Restricted pict (emf)  tool
Restricted pict (emf)  command  (Drawing menu)

Use this command to define a rectangular portion of the drawing sheet, and build an enhanced-metafile-image of the contents of this rectangle.

Operation: 	In one stage:

1st stage (selects):	Click somewhere, say A,...drag...release, say at B.

Comments:

The rectangular box defines the enclosing box of the enhanced-metafile-image.

You can cut-paste this image to the same or any other application that accepts enhanced-metafile-images.

Match points enabled:

No effect.


Drawing: Import pict  tool
Import picture  command  (Drawing menu)

Use this command to display a dialog, by means of which you navigate in folders and files and find a file in BMP or EMF format.This is then imported as a picture and becomes an object of your document.

Comments:

Another way to import such pictures in the drawing sheet, is via the clipboard, through copy-paste operations.


Drawing: Export pict  tool
Export picture  command  (Drawing menu)

Use this command to display a dialog, by means of which you navigate in folders and files and export a portion of your document as a BMP or EMF-format file with a name of your choice.

Before to use this command you must define some picture of this kind in your document and select it.

Comments:

This tool is more frequently used in combination with the Restricted pict (bmp) or Restricted pict (emf) tools.

Another way to export such pictures from you drawing sheet, is via the clipboard, through copy-paste operations, in the same or other documents.


Edit menu commands

The Edit menu offers the following commands:

Undo/Redo	Undo = Reverse previous editing operation. Redo = Undo last Undo!
Cut	Deletes data from the document and moves it to the clipboard.
Copy	Copies data from the document to the clipboard.
Paste	Pastes data from the clipboard into the document.
Select All	Selects all objects of the document.
Duplicate	Duplicates (exact copies) all selected objects, or sets the new translation vector through a predefined segment.
Rotate	Produces copies of selected objects rotated through some angle, or sets the new rotation center and rotation angle (through an angle).
Homothety	Produces copies of selected objects stretched/shrinked by some stretch-factor, or sets the new homothety-center and homothety-modulus through the data of a predefined segment.
Intersection	Finds the intersection points of two (previously selected) shapes.
Reflexion	Selects a line and then selects objects and reflects them on that line.
Inversion	Selects a circle and then selects objects (only segments/lines/circles transformed) and inverts them on that circle.
Translation	How to translate a single object or a group of selected objects.


Undo/Redo  command (Edit menu)

Use this command to reverse the last editing action, if possible.

Each undo can be followed by a redo.

Shortcuts

Keys:	CTRL+Z

Comments:

The application presently supports one-level-undo.

The menu-item and the tool-functionality alternate between undo and redo.

Each undo enables a redo and vice-versa.

So, don’t make more than one mistakes at a time, because you can correct only your last mistake.


Cut command (Edit menu)

Use this command to remove the currently selected data from the document and put it on the clipboard.

Shortcuts

Keys:	CTRL+X

Comments:

This command is unavailable if there is no data currently selected.

Cutting data to the clipboard replaces the contents previously stored there.


Copy command (Edit menu)

Use this command to copy selected data onto the clipboard.

Shortcuts

Keys:	CTRL+C

Comments:

This command is unavailable if there is no data currently selected.

Copying data to the clipboard replaces the contents previously stored there.


Paste command (Edit menu)

Use this command to insert a copy of the clipboard contents.

Shortcuts

Keys:	CTRL+V

Comments:

This command is unavailable if the clipboard is empty.

Pasting data from the clipboard, into the drawing-sheet, does not remove the clipboard contents.

You can paste objects copied or cutted from other applications of the following kinds

TEXT, BMP (bitmaps), EMF (enhanced-metafiles).

Additionally you can cut-copy-paste objects of the application, between the same or two different instances of the application, running on your computer.


Select All  command (Edit menu)

Use this command to select all contents of the drawing-sheet.

Shortcuts

Keys:	CTRL+A


Duplicate  command (Edit menu)

Use this command to produce exact duplicates of the selected objects.

Shortcuts

Mouse:	Click the toolbar-button

Keys:	CTRL+D

Comments:

The produced duplicates are shifted by a translation-vector, say (x,y), whose coordinates are given in cm and can be set by the appropriate fields in the Drawing Parameters dialog box, of the Settings menu.

You can set the translation-vector, through the data of a predefined segment as follows:

1) 	Select first a predefined segment (segment object, not a side of another shape).

2) 	Hold down the shift-key, while selecting this command from the menu or from the toolbar.

The translation-vector is set to the selected segment.


Rotate  command (Edit menu)

Use this command to produce rotated duplicates of the selected objects.

Use also this command to define new rotation center and rotation angle through the data of a predefined angle.

Shortcuts

Mouse:	Click the toolbar-button

Keys:	CTRL+R

Comments:

The produced duplicates are rotated about a center, say c=(x,y) (default: c=(0,0)),

and through an angle, say a (default: a=30 degrees).

You can change these data through the appropriate fields in the Drawing Parameters dialog box, of the Settings menu.

You can set the rotation-center and rotation-angle, through the data of a predefined angle as follows:

1) 	Select first a predefined angle (angle object, not an angle of another shape).

2) 	Hold down the shift-key, while selecting this command from the menu or the toolbar.

The rotation-center is set to the vertex of the selected angle.

The rotation-angle is set to the selected angle (oriented).


Homothety  command (Edit menu)

Use this command to produce homothetical duplicates of the selected objects.

Shortcuts

Mouse:	Click the toolbar-button

Keys:	CTRL+H

Comments:

The produced duplicates are transformed by homothety, with center say c=(x,y)  (default: c=(0,0)),

and through an expansion-factor, say a  (default: a=1.5 times).

You can change these data through the appropriate fields in Drawing Parameters dialog box, of the Settings menu.

You can set the homothety-center and homothety-modulus, through the data of a predefined segment as follows:

1) 	Select first a predefined segment (segment-object, not a side of another shape).

2) 	Hold down the shift-key, while selecting this command from the menu or from the toolbar.

The homothety-center is set to the first end-point of the selected segment.

The homothety-modulus is set to the length of the selected segment.


Intersection  command (Edit menu)

Use this command to get the intersection points of the first two selected objects.

Shortcuts

Mouse:	Click the toolbar-button

Comments:

Before to operate with this command, you have to select some objects.

For segments, this operation gives also intersection-points on their supporting line, beyond their end-points.

You can use this command, to visually solve a system of equations, through the intersection-points of the graphs of the functions, defining the equations.


Reflexion  command (Edit menu)

Use this command to select a line or segment or side of some polygon, say x, then, select some objects and reflect them on x (on the support-line of x).

Operation: 	In two stages:

1st stage (selects): 	Click near a line/segment/poly-side, say x, to select it.

2nd stage (selects): 	Click somewhere, say A,...drag...release, say at B.

Shortcuts

Mouse:	Click the toolbar-button

Comments:

The two stages of the tool are distinguished visually through the shape of the cursor.

In the first stage the cursor looks like    +R1.

In the second stage the cursor looks like    +R2.

The tool remains in its second stage and you can repeat the reflexion with other objects, with respect to the same line x.

To change to the first stage of the tool, hold down the shift-key, while you operate in the second stage.


Inversion  command (Edit menu)

Use this command to select a circle, say x, then, select some objects (actually only circles, lines and segments) and invert (or anti-invert) them on x.

Operation: 	In two stages:

1st stage (selects): 	Click near a circle, say x, to select it.

2nd stage (selects): 	Click somewhere, say A,...drag...release, say at B.

Shortcuts

Mouse:	Click the toolbar-button

Comments:

The two stages of the tool are distinguished visually through the shape of the cursor.

In the first stage the cursor looks like    +I1.

In the second stage the cursor looks like    +I2.

In the second stage, if, while selecting the object(s) to invert, you hold down the ctrl-key, the selected objects are anti-inverted with respect to x.

The tool remains in its second stage and you can repeat the inversion with other objects, with respect to the same circle x.

To change to the first stage of the tool, hold down the shift-key, while you operate in the second stage.

Actually, only segments, lines and circles are inverted.

Applied on segments, the iversion inverts their whole supporting line.


Translating objects

To translate a group of objects: Use the selection-tool to select the objects. Then, click on one of the selected objects and move the mouse, without releasing the button. Releas at the desired new position.

Using the keyboard only:

Select first some objects.

The group of selected objects is moved pixel-by-pixel, by pressing the corresponding arrow-keys.

Translate only one object + match some corner A of the translated object:

Click on the object near its corner A and drag it around.

While releasing, next to some other corner B (of another shape) hold down the ctrl-key.

The effect is that the moved object is translated so that its point A coincides with B.

For other options when translating objects look at Tiling the plane with objects.


Euclidean Geometry

This section of Help is still incomplete.

In this place we plan to present notes on E.G, developped in our courses, at the University of Crete. We hope to be able to present this material in the next update of this application.

As a sample we give an outline of the three first chapters: on history, on the axioms and on some elementary consequences of the axioms. Starting with the Elementary facts, the figures (and proofs) of the various subjects below appear in the corresponding sample-documents with the same name. F.e. all theorems formulated in “Elementary facts” are discussed again in the file “Elementary_facts” in the Samples folder.

Note on History	Notes on the history of Euclidean Geometry.
Axioms	The axioms of E.G. in the Hilbert-formulation.
Elementary facts	Theorems deriving directly from the axioms.
Areas	Theorems related to areas.
Ratio	Theorems related to ratio and proportion.
Triangles	Theorems on triangles.
Circles	Theorems on circles.
Quadrilaterals	Theorems on quadrilaterals.
Regular polygons	Theorems on regular polygons.
Bibliography on E. G	Commented bibliography on E.G..


Note on History (Euclidean Geometry)

The first systematic treatment of Euclidean Geometry are Euclid’s Elements(300 b.c). This, still modern treatise, establishes the style of mathematical writing: definitions-theorems-proofs. The work comprises 13 books, in which 465 theorems are proven. It starts with definitions and axioms, which are propositions accepted without proof (f.e. two points define exactly one line containing them). Then, various properties of the simplest shapes (triangles, polygons, circles) are proven, by reducing them to the axioms or, previously proven, consequences of them. Besides geometry, the work contains a treatise of proportions (book V) and number-theory (books VII, VIII, IX). Book X deals with irrationals and the last three books deal with solid geometry.

For about 2000 years, mathematics was nothing more than a discussion on the material of this work. A major issue on this discussion was the question of logical dependency of the axioms. The main question there was whether Euclid’s fifth postulate(axiom) depends logically on the other axioms and can be proven as a theorem. This axiom, essentially, says that from a point outside a line there is exactly one parallel to it. The independency of this axiom from the others was established by the work of Gauss, Bolyai and Lobatchefsky. They showed that by retaining all other axioms of Euclid’s Geometry and changing only the axiom on parallels, one can construct a new geometry (Hyperbolic Geometry of geometry of Bolyai-Lobatchefsky), as Euclid did, without ever ariving at a contradiction. This new geometry has a set of theorems in common with the Euclidean Geometry. Namely, the theorems that do not depend on the parallels-axiom. However, propositions that depend on that axiom are quite different in the two geometries. F.e.in the hyperbolic geometry the sum of the angles of a triangle can be any number x, with 0<x<pi.

A turning point uppon the way one looks at geometric problems was Descart’s (1596-1650) invention of coordinates. Using this idea one reduces geometric properties to algebraic equationns and relations between real numbers. This is important since it gives a general method to investigate every possible problem ariving in Euclidean Geometry. However the beauty of geometry is revealed mainly by the geometric reasoning and the discovery of the logical dependencies as Euclid teaches (synthetic method, based on the axioms). The fact that the science of mathematics starts with Euclid’s Geometry shows the dependency of mankind on Beauty. In the Greek language (Euclid’s language) the shape is called “scheme” and the word “ugly” is called “a-scheme” or non-shape.

Here my chaste Muse a liberty must take-

Start not! still chaster reader-she’ll be nice hence-

Forward, and there is no great cause to quake;

This liberty is a poetic licence,

Which some irregularity may make

In the design, and as I have a high sense

Of Aristotle and the Rules, ‘tis fit

To beg his pardon when I err a bit.

Byron, Don Juan, Canto I, 120


Axioms (Euclidean Geometry)

A thorough examination of the postulates (axioms) and the deductive reasoning applied in Euclid’s Elements, revealed several defects in its logical structure. In fact, for a solid foundation, uppon a complete set of axioms, one needs set-theory and (knowledge of the structure of the set of) real numbers (indeed, some of the modern axioms imply that the straight lines of Euclidean Geometry can be identified with the set of real numbers). These things weren’t clarified until the end of the 19th century. In these terms, the Euclidean plane can be considered as a set, say E. The various shapes are subsets of E. The elements of E are called points and we say that the point P is on the shape S, when the element P belongs to the subset S of E. Some shapes are more  important than others: Lines. The first group of axioms (formulated by Hilbert) describes some properties of lines:

Axiom 1.1 For every pair of points A, B of E, there is exactly one line L containing them.

Axiom 1.2 Every line L contains at least two distinct points, and there is at least one point not on the line.

Thus, every line has points on it and points not on it.

The next axioms complete the abstract description of lines by requiring that their points be well ordered. The essential of order is “betweeness”.

Axiom 2.1 If point C is between A and B, then all three points belong to the same line and C is also between B and A. Then, A is not between C and A and B is not between A and C.

Axiom 2.2 For any two distinct points A and B, there is a point C between A and B and there is also a point D, such that B is between A and D.

Axiom 2.3 For any three points A, B, C on a line, one of them is between the other two.

We see easily from these, that there are (infinite) many points between A, B and (infinite) many points not-between A, B (on both sides of the segment AB).

Next come the segments. These are defined by two points A, B of a line L, together with all points C, between them.

A triangle is defined by three distinct points A, B, C, not on a line and the segments that they define. A, B, C are the vertices. The segments c=AB, a=BC, b=CA are the sides of the triangle.

Axiom 2.4 A line intersecting one side of a triangle ABC and not passing through any of its vertices must intersect another side of the triangle.

i.e. If a line “enters” a triangle, crossing its border once, it must go out crossing a second time.

Next come axioms related to the “equality” between segments, angles and triangles. Segments can be identified with pairs of points (their end-points). The intuitive idea behind the term “congruent” is that shapes can be “equal” but lie at different places of the plane.

Axiom 3.1 For any two distinct points X, Y and a point A on a line L, there are exactly two distinct points B, C on L, such that (B,A) are congruent to (X,Y) and (A,C) is congruent to (X,Y), moreover A is between B and C.

Axiom 3.2 If two pairs of points (A,B), (C,D) are both congruent to a third pair, then they are also congruent to each other.

Axiom 3.3 if C is between A, B and C’ is between A’, B’ and (A,C) is congruent to (A’,C’) and (C,B) is congruent to (C’,B’) then also (A,B) is congruent to (A’,B’).

Thus, you can fix a point O on a line L and for every other segment (A,B), find a point X on L, such that (O,X) is congruent to (A,B). You can do this on both “sides” of O. In fact, a point O on the line L defines two subsets: L1, L2 the half-lines of L starting at O. These can be characterized by the fact that their union is the whole line, their intersection is {0} and for any pair of points (A,B) lying on Li, each point that lies between A and B lies also on Li (one says Li are convex sets).

After understanding half lines, you can define Angles as a pair of half-lines with a common start-point O. Then, with each triangle ABC you have three angles, which you can denote by the corresponding vertices A, B and C.

Axiom 3.4 If BAC is an angle whose sides do not lie in the same line and if A’, B’ are two distinct points, then there are exactly two half-lines A’C’ and A’C’’, such that the angles B’A’C’ and B’A’C’’ are both congruent to BAC, moreover if D’ is any point on the half line A’C’ and D’’ is any point on the half-line A’C’’, then the segment D’D’’ intersects the line A’B’.

Axiom 3.5 Every angle is congruent to itself.

Axiom 3.6 Let the triangle ABC have its sides b and c respectively congruent to the sides b’, c’ of the triangle A’B’C’ and the included angle A congruent to the included angle A’, then the other angles B, C are respectively congruent to the angles B’, C’.

In modern terms congruence is realized by isometric-transformations. These are maps of the plane on itself (F:E->E) that map any pair of points (A,B) to a congruent pair (A’,B’). The set of all such maps forms a Group and two shapes S, S’ are considered congruent, if there is an element F of G, such that F(S)=S’. The discovery of the intimate relationship of geometric equlaity of shapes (congruence) and group-theory stands behind the explosion of geometries (Klein).

One can invent various groups G, that “operate” on sets E, as groups of transformations and ask for properties of shapes that remain invariant under the transformations of G. All possible geometries are variations on this idea. The six axioms of congruence above, completely determine the isometric transformations of the Euclidean plane, by describing their properties as they act uppon segments, angles and triangles.

Next comes the famous parallels-axiom.

Axiom 4.1 Through a point A not lying on a given line L, passes exactly one parallel to L.

And last but not least the axioms that guarantee that lines are the “same” as the set R of real numbers:

Axiom 5.1 If B is between A and C, there is a positive integer n, such that repeating successively AB n-times on the same side from A, you pass beyond C. i.e. if B1=B, B2, ..., Bn are distinct to each other points on the same line with A, C and such that (A,B1), (B1,B2),...,(Bn-1,Bn) are all congruent, then it happens that C is between A and Bn.

Axiom 5.2 One cannot extend the set L of points of a line to a bigger set L’ without to violate some of the 9 axioms {1.1, 1.2, 2.1, 2.2, 2.3, 2.4, 3.1, 3.2, 5.1}.

Idealy, axiom 5.1 is valid in politics. You could start in a minor position, somewhere in the country, and after (sufficiently many) n steps become president of the U.S. Sadly that politics isn’t as well ordered as the set R of real numbers. Namely, axiom 5.2 appears not to be valid there. Some other guy extends L, and you remain behind, or you fall into a new hole of your old line.


Elementary facts (Euclidean Geometry)

All theorems of Euclidean Geometry are consequences of the axioms. One could think of something like the “logical distance” of a theorem from the axioms. There are the direct corollaries, which can be considered to be “next” to the axioms and the “heavy theorems” which need a long chain of logical steps, in order to deduce them from the axioms. In several places one has to define things (as in definitions-1,-2 below) and examine new properties.

Very close to the axioms are the theorems on congruent triangles.

Definition-1: Two triangles ABC and A’B’C’ are said to be congruent, when their corresponding sides are congruent and their corresponding angles are congruent too.

Theorem-1: (SAS) If the sides b, c of the triangle ABC are correspondingly congruent to the sides b’, c’ of A’B’C’ and the contained angle A is congruent to the corresponding contained angle A’, then the two triangles are congruent.

Theorem-2: (ASA) If a side a of the triangle ABC is congruent to a side a’ of A’B’C’ and the adjacent angles B, C are correspondingly congruent to B’, C’, then the two triangles are congruent.

Definition-2: A triangle ABC is said to be isosceles if it has two congruent sides. The remaining side is then called its base.

Theorem-3: The base-angles of an isosceles triangle are congruent and inversely: if the base angles of a triangle are congruent, then the triangle is an isosceles.

In the rest of the notes “congruent” is used interchangeably with “equal”. The following theorems can be considered to be close to the axioms. There are lots of such fine (and boring?) details. One has also to define such things as when a segment is greater than another, the interior and exterior domain of an angle,when an angle is greater than another, right-angles, right triangles, circles, rectangles, polygons etc.

Theorem-4: (SSS) If the sides a, b, c of the triangle ABC are correspondingly equal to the sides a’, b’, c’ of the triangle ABC, then the two triangles are equal.

Theorem-5: Each line L divides the plane in two parts (half-planes) L1, L2. If A is a point of L1 and B is a point of L2, then the segment AB intersects L in a point which is between A and B.

Theorem-6: For each segment AB, there is exactly one point M (the middle) such that M is between A and B and AM is equal to MB.

Theorem-7: If one side of a triangle is produced, the exterior angle is greater than either of the interior non-adjacent angles.

Theorem-8: In every triangle the sum of two of its angles is less than two right angles.

Theorem-9: In every triangle the greater angle is opposite to the greater side and inversely, the greater side is opposite to the greater angle.

Theorem-10: In a right triangle the hypotenuse is greater than either of the other sides.

Theorem-11: (Triangle-inequality) In every triangle the sum of two of its sides is greater than the third.

Theorem-12: A circle and a line have at most 2 points in common. If they have only one (tangent) then the radius to that point is orthogonal to the line.

Theorem-13: Through a point A outside a line L there is exactly one orthogonal to L.

Theorem-14: Through a point A on the line L there is exactly one orthogonal to L.

Theorem-15: If a line L cuts two other lines M, N so that the interior angles on one side of L have a sum less than two right angles, then M, N intersect on that side of L.

Theorem-16: If a line L cuts two parallel lines M, N, then it makes:

(i) alternate angles equal,

(ii) corresponding angles equal,

(iii) two interior angles on the same side of the line are supplementary.

Theorem-17 Two angles whose respective sides are parallel or orthogonal to one another are either equal or supplementary.

Theorem-18: The sum of the angles of a triangle is two right angles.

Theorem-19: The opposite sides and angles of a parallelogram are equal, each diagonal bisects the parallelogram, and the diagonals bisect one another.

Theorem-20: If three or more parallel lines cut equal segments on one trasversal, they cut equal segments on every other transversal.


Areas (Euclidean Geometry)

TODO


Ratio (Euclidean Geometry)

TODO


Triangles (Euclidean Geometry)

TODO


Cirles  (Euclidean Geometry)
Circles (Euclidean Geometry)

TODO


Quadrilaterals (Euclidean Geometry)

TODO


Regular polygons (Euclidean Geometry)

TODO


Bibliography (Euclidean Geometry)

TODO


File menu commands

The File menu offers the following commands:

New	Creates a new document.
Open	Opens an existing document.
Save	Saves an opened document using the same file name.
Save As	Saves an opened document to a specified file name.
Print	Prints a document.
Print Preview	Displays the document on the screen as it would appear printed.
Print Setup	Selects a printer and printer connection.
Exit	Exits the application.

Saving in other formats : You can save your work also in bmp and emf -formats.


File New command
New command (File menu)

Use this command to create a new document using the application.

You can open an existing document with the Open command.

Shortcuts

Keys:	CTRL+N


File Open command
Open command (File menu)

Use this command to open an existing document.

The application displays the File Open dialog box so you can select an existing file to open.

You can create new documents with the New command.

Shortcuts

Keys:	CTRL+O


File Save command
Save command (File menu)

Use this command to save the active document to its current name and directory.

When you save a document for the first time, the application  displays the Save As dialog box so you can name your document.

If you want to change the name and directory of an existing document before you save it, choose the Save As command.

The standard functioning of the command, described above applies only to the document-type iso, used by the application.

Saving in other formats : You can save your work also in two other standard windows-formats bmp and emf

Choosing this format, the application saves allways anew its content by creating files with the corresponding user-supplied name and the appropriate extension.

Shortcuts

Keys:	CTRL+S


File Save As command
Save As command (File menu)

Use this command to save and name the active document.

The application  displays the Save As dialog box so you can name your document.

To save a document with its existing name and directory, use the Save

Saving in other formats does not automatically saves also in the proprietary type iso.


Saving in other formats
Saving with caution

After you save the content of a document in the form of bmp or emf, you should take care of the contents of the document also in the form iso. Unless you are decided not to save your work in iso-format, evoke a second time the dialog and save the document in its proprietary format.

The two other types are included, in order to be able to import documents, produced by the application, to other documents, which do not understand the proprietary form iso, but can work with bmp and/or  emf .

You can also transfer data from/to other applications via the clippboard in combinations with Restricted pict (bmp) to isolate a portion of the drawing and produce a bitmap-object on the document, or with Restricted pict (emf) to isolate a portion of the drawing and produce an enhanced metafile-object on the document.


File 1, 2, 3, 4 command
1, 2, 3, 4 command (File menu)

Use the numbers and filenames listed at the bottom of the File menu to open the last four documents you closed.  Choose the number that corresponds with the document you want to open.


File Exit command
Exit command (File menu)

Use this command to end your Isoptikon-session.

You can also use the Close-button

on the top-right corner of the application’s window.

The application  prompts you to save documents with unsaved changes.

Shortcuts

Mouse:	Click the application's close-button.

Keys:	ALT+F4


File Open dialog box

The following options allow you to specify which file to open:

File Name

Type or select the filename you want to open.  This box lists files with the extension you select in the List Files of Type box.

List Files of Type

Select the type of file you want to open:

The appication’s file type is iso.

You can also import pictures stored in files of type  bmp  and  emf  (these are respectively bitmaps and enhanced metafiles).

Drives

Select the drive in which the application  stores the file that you want to open.

Directories

Select the directory in which the application  stores the file that you want to open.

Network...

Choose this button to connect to a network location, assigning it a new drive letter.


File Save As dialog box

The following options allow you to specify the name and location of the file you're about to save:

File Name

Type a new filename to save a document with a different name.  A filename can contain up to eight characters and an extension of up to three characters.

The application  adds the extension you specify in the Save File As Type box.

List Files of Type

Select the type of file you want to create:

The appication’s file type is iso.

You can also save pictures in files of type bmp and emf (these are respectively bitmaps and enhanced metafiles).

Drives

Select the drive in which you want to store the document.

Directories

Select the directory in which you want to store the document.

Network...

Choose this button to connect to a network location, assigning it a new drive letter.


Help menu commands

The Help menu offers the following commands, which provide you assistance with this application:

Help Topics	Use this command to display the opening screen of Help.   From the opening screen, you can jump to step-by-step instructions for using the application  and various types of reference information.   Once you open Help, you can click the Contents button whenever you want to return to the opening screen.
Help Index	Use this command to display the index of help topics of the application.
Theory	Jumps to the text exposing the theory of isoptic cubics.
About	Displays the version number of this application.


Help Using Help Command
Context Help command

Use the Context Help command to obtain help on some portion of the application  When you hold down the shift-key and press the F1-key, the mouse pointer will change to an arrow and question mark.  Then click somewhere in the application’s  window, such as another Toolbar button.  The Help topic will be shown for the item you clicked.

Shortcut

Keys:	SHIFT+F1


About command (Help menu)

Use this command to display the copyright notice and version number of your copy of the application.


Help index  command (Help menu)

Use this command to display the index of the help-contents of  the application.


No Help Available

No help is available for this area of the window.


Title Bar

Isoptikon

The title bar is located along the top of a window.  It contains the name of the application and document.

To move the window, drag the title bar.  Note: You can also move dialog boxes by dragging their title bars.

A title bar may contain the following elements:

Application Control-menu button

Document Control-menu button

Maximize button

Minimize button

Name of the application

Name of the document

Restore button


Scroll bars

Displayed at the right and bottom edges of the document window.  The scroll boxes inside the scroll bars indicate your vertical and horizontal location in the document.  You can use the mouse to scroll to other parts of the document.


Theory

A more or less complete exposition of the theory of isoptic cubics.


Why use Isoptikon
Why use the application

Because you are an aesthete (of course).

Because you can easily produce high precision drawings of Euclidean Geometry.

Because the exact drawing of figures becomes indispensable as the theorems (and their corresponding figures) become more and more complex (look f.e. at some samples included in the application’s folder).

Because sometimes exact drawing suggests a new theorem.

Because you can draw isoptic cubics, and thus, realize their beauty and their abundant geometric properties.

The application was devoted initially to the study of these cubics.

Then it evolved to an instrument of Euclidean Geometry.

Of course, there are plenty of similar tools out there (on Euclidean geometry actually, not on isoptic cubics). But they have two main drawbacks:

You have to learn a lot in order to use them (whereas small is beautiful).

You have to pay ad infinitum, to by them and their updates.

Here we try to realize these three concepts: Quality, Simplicity, Availability.

In fact, the application was several times used in lectures on Euclidean Geometry, in lectures on hyperbolic geometry, and was proven to be able to draw, within minutes, every figure needed.

When we realized this, we stopped developing further tools for the application and we gave it to our students to use it and enjoy the construction of fine drawings ... plus the bonus of insight into the shapes of isoptic cubics. This is a start ...

Of course, in order to play with this application:

A minimum knowledge of Euclidean Geometry is required.

But.

To read the chapters on isoptic cubics a mathematical background is required. These chapters and the corresponding tools for these cubics are intendent for the mathematician, who is interested on this kind of curves.

That is the usual method, but not mine-

My way is to begin with the beginning;

The regularity of my design

Forbids all wandering as the worst of sinning,

And therefore I shall open with a line

(Although it cost me half an hour in spinning)

Narrating somewhat of Don Juan’s father,

And also of his mother, if you’d rather.

Byron, Don Juan, Canto I, 7


How to use Isoptikon

How to use this application

Create documents with the extension  iso

This is the default “proprietary” format of the application.

The documents can contain geometric shapes, text boxes, bitmaps and enhanced metafiles. Bitmaps and enhanced metafiles can be constructed with the application or imported in documents of the application, via the clipboard or the open-file-dialog. In the later cases, they are considered as objects and can be moved, modified etc. as any other object of the document.

The various drawing-tools are grouped together in the Drawing menu and its submenus:

To produce drawings of Euclidean Geometry, use the tools in the following menus:

Segments menu

Lines menu

Triangles menu

Parallelograms menu

Angles menu

Circular Arcs menu

Circles menu

To produce drawings of isotpic-cubics, use the tools contained in the Isoptics menu.

To learn about isoptic-cubics look at the topics Apollonian (isoptic) cubics.

Select items from these menus (or use the shortcuts in the toolbars), draw shapes and save your document with the default extension iso.

You have full control of the sheet-dimensions, axes location, unit lengths etc.

To modify these settings, use the Drawing sheet  commmand from the Settings menu .

To learn quickly to use the application, you can start with the tutorials and then start producing your own documents or modify some of the included samples.

Create documents of bitmap-format (extension BMP)

Create a new document, draw on it, using various tools and save the document with the extension BMP.

Because such files consume huge amounts of memory, there is a restriction on the dimensions of the drawing sheet, when producing files of this kind.

Your sheet dimensions are not allowed to exceed the standard A4-dimensions, which are 21x29.6 cm.

Often the interesting part of the drawing is smaller than the whole sheet containing it.

Thus, it is often preferable to use the tool Restricted pict (bmp) to isolate a portion of the drawing and produce a bitmap-object on the document, which subsequently can be copied and pasted to other documents.

You can also use the previous tool in combination with the Export picture tool, to save such a document-portion in a single file, with a name of your choice.

Create documents of Enhanced-metafile-format (extension EMF)

Create a new document, draw on it, using various tools and save the document with the extension EMF.

Often the interesting part of the drawing is smaller than the whole sheet containing it. Thus, it is often preferable to use the tool Restricted pict (emf) to isolate a portion of the drawing and produce an enhanced metafile-object on the document, which subsequently can be copied and pasted to other documents.

If you need an EMF-type-file containing exactly this interesting part of the document, you can delete all other contents and reduce the sheet-size to match the dimensions of the picture created by the above method. Then save your document with the extension EMF.

It is often preferable to use the tool Restricted pict (emf) to isolate a portion of the drawing and produce an enhanced-metafile-object on the document, which subsequently can be copied and pasted to other documents.

You can also use the previous tool in combination with the Export picture tool, to save such a document-portion in a single file, with a name of your choice.


Exact drawing

The main problems for exact drawing in geometry are:

a)   To match a point, f.e. one vertex of a triangle, with another existing point,

b)   To put a point exactly on a shape, f.e. on the side of an existing triangle,

c)   To determine the intersection of two shapes, f.e. of two circles,

d)   To draw parallels, orthogonals etc. to selected segments, from given points,

e)   To produce exact or similar copies of shapes.

To solve the first problem, the application uses the tool Match points , by means of which a new point is forced to coincide with another, lying sufficiently close to it. The “sensitivity” or how far from the old point you must click in order to achieve coincidence, is set by the appropriate field of the Drawing Parameters... dialog.

To solve the second and third problem, the application uses the tool Set points , by means of which (if you hold down the shift-key, when clicking) the application forces the point to be set on the next lying side of the next lying shape. The same tool (holding down now the ctrl-key) forces the point to coincide with the next intersection point of the two next lying shapes.

The fourth problem is solved by specialized drawing-tools, for segments, lines, circles, which operate in two stages. In the first stage you select f.e. a circle, while in the second you draw, say, a new circle, orthogonal to the one selected before.

The last problem is solved by implementing the “reflections”, “rotations”, “translations” and “similarities” of the Euclidean Geometry.

Of course one can use the circle tool in the place of the traditional compass, the line-tool in the place of rule in combination with the set-points tool.


Isoptics  menu commands

The Isoptics menu contains all the tools needed for drawing Isoptic Cubics and related to them shapes.

Define through ABCD	Use this tool to define four points ABCD, the corresponding segments AB, CD and the isoptic cubic AP(AD,BC).
Through predefined segments	Use this tool to select two, previously defined segments AB, CD, and get the isoptic AP(AD,BC).
Through predefined angle	Use this tool to select a, previously defined angle, and get the corresponding Singular isoptic.
Singular	Use this tool to define an angle and the related singular isoptic.
Through constants	Displays a dialog, which enables the definition of an isoptic through its canonical constants.
Isoptic-II	Use this tool to select a, previously defined, isoptic cubic and get the associated to it isoptic-II.
Isoptic-III	Use this tool to select a, previously defined, isoptic cubic and get the associated to it isoptic-III.
Canonical axes	Use this tool to select a, previously defined, isoptic cubic and get its associated canonical axes.
Asymptotic	Use this tool to select a, previously defined, isoptic cubic and get its associated Asymptotic.
Middle line	Use this tool to select a, previously defined, isoptic cubic and get its associated middle-line.
Flexes	Use this tool to select a, previously defined, isoptic cubic and get its associated flexes.
Group operation	A composite tool operating in several stages, to visualize the group structure of an isoptic cubic.
Main involution	Tool operating in two stages, to visualize the main involution of an isoptic cubic.
Other inversions	Tool operating in three stages, to visualize the invariant inversions of an isoptic cubic.
Angle animation	Tool operating in two stages and used to visualize the angles from points of the isoptic cubic, to its defining segments AB, CD.
Half-component pts	Displays a dialog for setting the half-number of points of an isoptic-cubic-component.


Define through ABCD  command (Isoptics menu)

Use this command to create a new isoptic AP(AD,BC) by defining two segments, x=AB and y=CD.

Operation: 	In two stages:

1st stage (draws): 	Click somewhere, say A,...drag...release, say at B.

2nd stage (draws): 	Click somewhere, say C,...drag...release, say at D.

Comments:

In the first stage you define a segment x=AB. All rules for segment-drawing apply here.

In the second stage you define a segment y=CD. All rules for segment-drawing apply here.

The application inserts the isoptic cubic AP(AD,BC).

Match-points enabled:

If pre-existing points nearby, readjustement of all A, B, C and D.


Through predefined segments  command (Isoptics menu)

Use this command to create a new isoptic AP(AD,BC), by selecting two existing segments, x=AB and y=CD.

Operation: 	In two stages:

1st stage (selects): 	Click somewhere to select a segment, say x.

2nd stage (selects): 	Click somewhere to select a segment, say y.

Comments:

The application inserts the isoptic AP(AD,BC).

This is useful if you want to draw specialized or degenerate isoptics.

Match-points enabled:

No-effect.


Through predefined angle  command (Isoptics menu)

Use this command to construct a singular isoptic through an existing angle.

Operation: 	In one stage:

1st stage (selects): 	Click somewhere to select an angle, say x.

Comments:

The application inserts the singular isoptic, corresponding to x.

Match-points enabled:

No-effect.


Singular  command (Isoptics menu)

Use this command to construct a singular isoptic, by defining an angle, say x.

Operation: 	In two stages:

1st stage (draws): 	Click somewhere, say A,...drag...release, say at B.

2nd stage (draws): 	Click somewhere, say B1,...drag...release, say at C.

Comments:

You actually construct an angle. All rules for angle-construction apply here.

The application supplies the corresponding singular isoptic.

Match-points enabled:

If pre-existing points nearby, readjustement of all three A, B and C.


Through constants  command (Isoptics menu)

Use this command to get a dialog-box, where you can set the canonical constants of an isoptic.

The application displays a dialog with three fields for the constants a, b, c entering in the formula, defining the canonical form.

You can define also singular isoptic cubics, by entering values only for a and c (in that case b depends on a, c) and pressing the button “SINGULAR”.

Comments:

The application calculates and draws the corresponding isoptic, using as canonical constants the x,y-axes of the drawing sheet.


Isoptic-II  command (Isoptics menu)
Isoptic II  command (Isoptics menu)

Use this tool to select a, previously defined, isoptic cubic and get the associated to it isoptic-II cubic.

Operation: 	In one stage:

1st stage (selects): 	Click somewhere to select an isoptic, say x.

Comments:

The application inserts the Isoptic-II, isoptic of the selected one.


Isoptic-III  command (Isoptics menu)
Isoptic III  command (Isoptics menu)

Use this tool to select a, previously defined, isoptic cubic and get the associated to it isoptic-III cubic.

Operation: 	In one stage:

1st stage (selects): 	Click somewhere to select an isoptic, say x.

Comments:

The application inserts the Isoptic-III, isoptic of the selected one.


Canonical axes  command (Isoptics menu)

Use this command to select a, previously defined, isoptic cubic and get its associated canonical axes.

Operation: 	In one stage:

1st stage (selects): 	Click somewhere to select an isoptic, say x.

Comments:

The application inserts the canonical axes of the selected isoptic.


Asymptotic  command (Isoptics menu)

Use this command to select a, previously defined, isoptic cubic and get its asymptotic line.

Operation: 	In one stage:

1st stage (selects): 	Click somewhere to select an isoptic, say x.

Comments:

The application inserts the asymptotic line of the selected isoptic.


Middle line  command (Isoptics menu)

Use this command to select a, previously defined, isoptic cubic and get its middle-line.

Operation: 	In one stage:

1st stage (selects): 	Click somewhere to select an isoptic, say x.

Comments:

The application inserts the middle line of the selected isoptic.


Flexes  command (Isoptics menu)

Use this command to select a, previously defined, isoptic cubic and get its flex-points.

Operation: 	In one stage:

1st stage (selects): 	Click somewhere to select an isoptic, say x.

Comments:

The application inserts the one/three flexes of the selected isoptic.


Group operation  command (Isoptics menu)

Use this command to select a, previously defined, isoptic cubic and operate manually with its group structure.

Operation: 	In an arbitrary number k>2 stages:

1st stage (selects): 	Click somewhere to select an isoptic, say x.

2nd stage (selects): 	Click somewhere to select the origin, say O, of x.

3rd stage (set point): 	Click somewhere to set a point, say A(1), of x.

4rth stage (set point): 	Click somewhere to set a point, say B(1), of x.

................................................

k=(2n+1)-stage (set point)	Click somewhere to set a point, say A(n), of x.

k=(2n+2)-stage (set point)	Click somewhere to set a point, say B(n), of x.

................................................

Comments:

In the first stage you select an existing isoptic cubic, say x.

In the second stage, click near x to select the next lying flex-point as origin.

If in the second stage, you hold down the shift-key, then instead of a flex, the application sets the origin O to be the next point on x, of your click-point.

Starting from the third stage, you set pairs of points A(n), B(n), by clicking near x.

Then, the application calculates and draws the resulting C(n)=A(n)+B(n), where “+” is the group operation of the isoptic cubic.

If in the third and subsequent stages you hold down the ctrl-key, while you click, the tool returns to its first stage and you can start anew by selecting another isoptic cubic.


Main involution  command (Isoptics menu)

Use this command to select a, previously defined, isoptic cubic and visualize its main involution.

Operation: 	In two stages:

1st stage (selects): 	Click somewhere to select an isoptic, say x.

2nd stage (draws): 	Click somewhere, say A,...drag...release, say at B.

Comments:

In the first stage the user selects a predefined isoptic cubic.

Then, the application calculates and draws its main-involution-circle.

Subsequent clicks define pairs of conjugate points with respect to this circle.

If, in the second stage, the shift-key is hold down, the application finds the next to the click-point lying on the curve and its conjugate point (lying also on the curve).

The segments joining these points to the center of the main-involution-circle are also drawn.


Other inversions  command (Isoptics menu)

Use this command to select a, previously defined, isoptic cubic and visualize it invariant inversions.

Operation: 	In three stages:

1st stage (selects): 	Click somewhere to select an isoptic, say x.

2nd stage (selects): 	Click somewhere, say A, to select a circle, say x.

3rd stage (draws): 	Click somewhere, say B,...drag...release, say at C.

Comments:

In the first stage the user selects a predefined isoptic cubic, say x.

The application supplies the two/four inversion circles.of the isoptic x.

In the second stage, the user selects the inversion circle, say y, whose circumference is next to the click-point A.

In the third stage the user draws C and the application supplies the inversed point D of C.

It draws also the radii, joining C, D with the center of y.

If, in the third stage, the shift-key is hold down, the application finds the next to the click-point lying on the curve and its inversed point (lying also on the curve).

If, in the third stage, the ctrl-key is hold down, the tool turns to its first stage.


Angle animation  command (Isoptics menu)

Use this command to select a, previously defined, isoptic cubic and visualize its main property of viewing two segments under equal angles.

Operation: 	In two stages:

1st stage (selects): 	Click somewhere to select an isoptic, say x.

2nd stage (draws): 	Click somewhere, say E,...drag...release, say at F.

Comments:

In the first stage the user selects a predefined isoptic cubic, say x.

In the second stage the application determines the point G, on x, lying next to F, and supplies the segments GA, GB, GC, GD, where AB, CD are two segments defining the isoptic cubic.


Half-component-pts  dialog (Isoptics menu)
Half-component-pts  command (Isoptics menu)

Displays a dialog-box, where you can set the half-number N of component-points of isoptic-cubics.

The application uses this number as a maximum. It ignores points that are too far out of the actual drawing sheet. Since isoptics have at most two components, such a cubic carries at most 4N points.


Lines  menu commands

The Line menu offers the following commands, relateted to the construction of lines:

Simple line	Tool to draw a line passing through two points. The line extends to the boundary of the drawing-sheet.
Orthogonal to_	Tool to draw a line orthogonal to another segment or line or side of some polygon. The tool operates in two stages.
Parallel to _	Tool to draw a line parallel to another segment or line or side of some polygon. The tool operates in two stages.
Middle orthogonal to_	Tool to draw a line orthogonal in the middle of another segment or side of some polygon.
Tangent at_	Tool to draw a line, tangent to a circle, defined previously, at some point of it.
Tangent to _	Tool to draw two lines tangent to a circle, defined previously.
Common tangents to 2 circles	Tool to draw the common tangent lines to two predefined circles.


Simple line  command (Lines menu)

Use this command to create a new line by defining two of its points.

Operation: 	Click somewhere, say at A,...drag...release, say at B.

The application inserts the line through A, B.

The line extends to the boundary of the drawing-sheet.

Shortcuts

mouse	click the line-tool on the toolbar

Comments:

Holding down the cttl-key, while defining the line, forces it to be horizontal/vertical/45-slope, depending of the direction of the movement.

Match-points enabled:

If pre-existing points nearby, say X near A, Y near B, readjustement of both A and B. Thus, to lay a line, passing through the existing points X and Y, click next to X and release next to Y.


Parallel to  command (Lines menu)

Use this command to construct a line parallel to a pre-defined segment or line or side of some polygon, say x.

Operation: 	In two stages:

1st stage (selects): 	Click somewhere to select a segment/line/side, say x.

2nd stage (draws): 	Click somewhere, say A,...drag...release, say at B.

Comments:

The application forces the line to be parallel to x, and pass through A.

The tool continues with its second stage: You can continue to draw parallel to the same object, selected in the first stage.

You can turn it to its first stage (in order to select another x) by holding down the ctrl-key, while you release the mouse (defining B).

Match-points enabled:

If pre-existing points nearby, readjustement of A only. Thus, you can lay with precision a parallel to a given line, passing through an existing point A.


Orthogonal to  command (Lines menu)
Othogonal to  (Lines menu)

Use this command to construct a line orthogonal to an existing segment/line/polygon-side, say x.

Operation: 	In two stages:

1st stage (selects): 	Click somewhere to select a segment/line/side, say x.

2nd stage (draws): 	Click somewhere, say A,...drag...release, say at B.

Comments:

The application forces the line to be orthogonal to x, and pass through B.

The tool continues with its second stage: You can continue to draw lines, orthogonal to the same object, x, selected in the first stage.

You can turn it to its first stage (in order to select another x) by holding down the ctrl-key, while you release the mouse (defining B).

Match-points enabled:

If pre-existing points nearby, readjustement only of B. Thus you can lay with precision a line orthogonal to x passing through an existing point B.


Tangent to  command (Lines menu)

Use this command to construct the two tangent lines to an existing circle, say x, from a point, say B.

Operation: 	In two stages:

1st stage (selects): 	Click somewhere to select a circle, say x.

2nd stage (draws): 	Click somewhere, say A,...drag...release, say at B.

Comments:

The application inserts two lines from B, tangent to x, if A, B are outside the circle x.

The tool continues with its second stage: You can continue to draw tangents to the same circle x, selected in the first stage.

You can turn it to its first stage (in order to select another x) by holding down the ctrl-key, while you release the mouse (defining B).

Match-points enabled:

If pre-existing points nearby, readjustement of B.


Tangent at  command (Lines menu)

Use this command to construct a line, tangent to an existing circle, at a point of it.

Operation: 	In two stages:

1st stage (selects): 	Click somewhere to select a circle, say x.

2nd stage (draws): 	Click somewhere (say A)...drag...release (say at B).

x is the selected circle.

The application calculates the point C on x, lying next to B, and draws there the tangent-line to x.

Comments:

The tool continues with its second stage: You can continue to draw tangents at the same object x, selected in the first stage.

You can turn it to its first stage (in order to select another x) by holding down the ctrl-key, while you release the mouse (defining B).

Match-points enabled:

If pre-existing points on x nearby B, readjustement of B.

Thus, you can draw a line tangent to x at some pre-defined point of x, say C, by clicking near C.


Middle orthogonal to  command (Lines menu)

Use this command to construct the middle-orthogonal line of an existing segment, say x.

Operation: 	1 stage (selects): 	Click somewhere to select a circle, say x.

The application inserts the middle-orthogonal of x.

Comments:

You can continue to select segments and construct their middle-orthogonals.

Match-points enabled:

No-effect.


Common tangents to_2  command (Lines menu)

Use this command to construct the common tangents (four) of two existing circles.

Operation: 	In two stages:

1st stage (selects): 	Click somewhere to select a circle, say x.

2nd stage (selects): 	Click somewhere to select a circle, say y.

x, y are the selected circles.

The application inserts the (four in general) common tangent lines of x, y.

Comments:

The tool returns to its first stage: You can continue to select pairs of circles, and draw their common tangent lines.

Match-points enabled:

No-effect.


Parallelograms  menu commands

The Parallelograms menu offers the following commands, relateted to the construction of parallelograms:

General	Tool used to draw parallelogramms or rectangles.
Screen	Tool used to draw rectangles with sides parallel to the screen-sides.
Equilateral	Tool used to draw equilateral parallelograms (rhombii).
Trapezium	Tool used to draw equilateral trapezii.


General parallelogram  command (Parallelograms menu)

General  command (Parallelograms menu)

Use this command to create a new parallelogram, with base-side, say x=AB, and opposite to x side, say y=CD.

Operation: 	In two stages:

1st stage (draws): 	Click somewhere, say A,...drag...release, say at B.

2nd stage (draws): 	Click somewhere, say B1,...drag...release, say at C.

Shortcuts

mouse	click the parallelogram-tool on the toolbar

Comments:

The first stage produces a segment x=AB. All rules for segment-construction apply here.

In the second stage, holding down the cttl-key, produces a rectangle, based on x=AB.

In th second stage, the application completes the base-segment x, using the point C. It inserts a segment CD parallel and equal to AB, such that ABCD is positively (counterclockwise) oriented.

Match-points enabled:

If pre-existing points nearby, readjustement of all three A, B and C.


Screen  command (Parallelograms menu)

Use this command to create a parallelogram, whose sides are parallel to the screen-sides. The parallelogram, say x, is determined by its top-left vertex, say A, and its bottom-right vertex, say B.

Operation: 	In one stage:

1st stage (draws): 	Click somewhere, say A,...drag...release, say at B.

Comments:

The application inserts the rectangle, having A and B as its top-left, bottom-right vertices, respectively.

Match-points enabled:

If pre-existing points nearby, readjustement of both A and B.


Equilateral  command (Parallelograms menu)

Use this command to create a rhombus (equilateral parallelogram), say x=ABCD.

Operation: 	In two stages:

1st stage (draws): 	Click somewhere, say A,...drag...release, say at B.

2nd stage (draws): 	Click somewhere, say B1,...drag...release, say at C.

Comments:

The first stage produces a segment x=AB. All rules for segment-construction apply here.

In the second stage, holding down the cttl-key, produces a square, based on x=AB.

In th second stage, the application completes the base-segment x, using the point C. It inserts a segment CD parallel and equal to AB, which is modified, so that AB=BC=CD.

Match-points enabled:

If pre-existing points nearby, readjustement of A and C.


Trapezium  command (Parallelograms menu)

Use this command to construct an equilateral trapezium.

Operation: 	In two stages:

1st stage (draws): 	Click somewhere, say A,...drag...release, say at B.

2nd stage (draws): 	Click somewhere, say B1,...drag...release, say at C.

Comments:

The first stage produces a segment x=AB. All rules for segment-construction apply here.

In th second stage, the application completes the construction by supplying the opposite to x parallel y= CD, such that BC=DA.

Match-points enabled:

If pre-existing points nearby, readjustement of A, B and C.


Segments menu commands

The Segments menu offers the following commands, relateted to the construction of line-segments:

Simple segment	Tool to draw a line-segment joining its end-points.
Cross	Tool to draw a line-segment joining its end-points. The application adds two segments orthogonal on the middle of the one, drawn by the user.
Ratio	Tool to draw a line-segment joining its end-points. The application divides the segment, drawn by the user, in two parts whose lengths are in a given ratio.
Parallel to _	Tool to draw a line-segment parallel to another segment or line or side of some polygon.
Orthogonal to _	Tool to draw a line-segment orthogonal to another segment or line or side of some polygon.
Tangent to _	Tool to draw two line-segments tangent to a circle, defined previously.
Tangent at _	Tool to draw a line-segment, tangent to a circle, defined previously, at some point of it.
Common tangents to _2 _	Tool to draw the common tangents to two predefined circles. The tool acts as a selection tool.
Complete quadrilateral to _	Tool to draw the complete quadrilateral of two predefined segments. The tool acts as a selection tool.


Simple segment  command (Segments menu)

Use this command to create a new line-segment.

Operation: 	Click somewhere, say at A,...drag...release, say at B.

A is the first end. B is the second end of the segment AB.

Shortcuts

mouse	click the segment-tool on the toolbar

Comments:

Holding down the cttl-key, while defining the segment, forces it to be horizontal/vertical/45-slope, depending of the direction of the movement.

Match-points enabled:

If pre-existing points nearby, readjustement of both A and B.


Cross  command (Segments menu)

Use this command to construct a segment, say x, together with one other segment, say y, orthogonal to x, in its middle-point

Operation: 	Click somewhere, say A,...drag...release, say at B.

Comments:

You draw x. Then y, are supplied by the application.

Use this tool if the middle-point of the segment is important in your construction.

Holding down the cttl-key, while defining the segment, forces it to be horizontal/vertical/45-slope, depending of the direction of the movement.

Shortcuts

mouse	click the button on the toolbar

Match-points enabled:

If pre-existing points nearby, readjustement of both A and B.


Ratio  command (Segments menu)

Use this command to construct two consecutive segments, AB and BC, of a given ratio k.

|AB|/|BC| = k.

Operation:	Click somewhere, say A,...drag...release, say at C.

Comments:

The application inserts a point B and divides the segment in two parts AB, BC so that |AB|/|BC| = k. In total 3 segments are produced: AB, BC and BD equal to AB and orthogonal to it.

The ratio k can be set through the appropriate field in the Drawing Parameters dialog.

Holding down the shift-key, while defining the segment, forces it to be horizontal/vertical/45-slope, depending of the direction of the movement.

Holding down the ctrl-key, while defining the segment, divides it to the golden-ratio = 0.618...

Match-points enabled:

If pre-existing points nearby, readjustement of both A and C.


Parallel to  command (Segments menu)

Use this command to construct a segment parallel to a pre-defined segment or line or side of some polygon, say x.

Operation: 	In two stages:

1st stage (selects): 	Click somewhere to select a segment/line/side, say x.

2nd stage (draws): 	Click somewhere, say A,...drag...release, say at B.

Comments:

The application forces the segment to be parallel to x, and pass through A.

The tool continues with its second stage: You can continue to draw parallel to the same object, selected in the first stage.

You can turn it to its first stage (in order to select another x) by holding down the ctrl-key, while you release the mouse (defining B).

You can use this command to extend a given segment or side of polygon to an arbitrary length.

To extend a segment by its very same length on its left or on its right (doubling it), select it first and then, holding down the shift-key, press the left or right-arrow key.

Match-points enabled:

If pre-existing points nearby, readjustement of A only.


Orthogonal to  command (Segments menu)
Othogonal to  (Segments menu)

Use this command to construct a segment orthogonal to an existing segment/line/polygon-side, say x.

a) From a given point outside the segment x.

b) From a given point on the segment x.

Operation: 	In two stages:

1st stage (selects): 	Click somewhere to select a segment/line/side, say x.

2nd stage (draws): 	Click somewhere, say A,...drag...release, say at B.

Shortcuts

mouse	click the button on the toolbar

Comments:

The application forces the segment to be orthogonal to x, and pass through B. This is the normal construction, used to define a segment orthogonal to x, from a point B outside x.

If you have a point A on x and want to draw from A, orthogonal to x, proceed as follows:

Activate first the match-points tool.

Then hold down the shift key and click near A. Drag ... release somewhere in B (not near an existing point). Release the shift-key.

The tool continues with its second stage: You can continue to draw segments, orthogonal to the same object, x, selected in the first stage.

You can turn it to its first stage (in order to select another x) by holding down the ctrl-key, while you release the mouse (defining B).

Match-points enabled:

If pre-existing points nearby, readjustement only of B.


Tangent to  command (Segments menu)

Use this command to construct the two tangent segments to an existing circle, say x..

Operation: 	In two stages:

1st stage (selects): 	Click somewhere to select a circle, say x.

2nd stage (draws): 	Click somewhere, say A,...drag...release, say at B.

Comments:

The application inserts two segments from B, tangent to x, if A, B are outside the circle x.

The tool continues with its second stage: You can continue to draw tangents to the same circle x, selected in the first stage.

You can turn it to its first stage (in order to select another x) by holding down the ctrl-key, while you release the mouse (defining B).

Match-points enabled:

If pre-existing points nearby, readjustement of B.


Tangent at  command (Segments menu)

Use this command to construct a segment, tangent to an existing circle, at a point of it.

Operation: 	In two stages:

1st stage (selects): 	Click somewhere to select a circle, say x.

2nd stage (draws): 	Click somewhere (say A)...drag...release (say at B).

x is the selected circle.

The application calculates the point C on x, lying next to B, and draws there the tangent to x.

Comments:

The tool continues with its second stage: You can continue to draw tangents at the same object x, selected in the first stage.

You can turn it to its first stage (in order to select another x) by holding down the ctrl-key, while you release the mouse (defining B).

Match-points enabled:

If pre-existing points on x nearby B, readjustement of B.

Thus, you can draw a segment tangent to x at some pre-defined point of x, say C, by clicking near C.


Common tangents to_2  command (Segments menu)

Use this command to construct the common tangents (four) of two existing circles.

Operation: 	In two stages:

1st stage (selects): 	Click somewhere to select a circle, say x.

2nd stage (selects): 	Click somewhere to select a circle, say y.

x, y are the selected circles.

The application inserts the (four in general) common tangents of x, y.

Comments:

The tool returns to its first stage: You can continue to select pairs of circles, and draw their common tangents.

Match-points enabled:

No-effect.


Complete quadrilateral  command (Segments menu)
Complete quadrilateral  (Segments menu)

Use this command to construct the complete quadrilateral of two existing segments.

Operation: 	In two stages:

1st stage (selects): 	Click somewhere to select a segment, say x.

2nd stage (selects): 	Click somewhere to select a segment, say y.

The application inserts a triangle (3 segments rather) whose vertices define the other 3 missing (besides the end-points of x and y) points of the corresponding complete quadrilateral.

Comments:

The tool returns to its first stage. Continue by selecting a new pair.

Match-points enabled:

No-effect.


Settings  menu commands

The settings menu contains items which adjust/customize the drawing sheet and other parameters involved in the various tools of the application.

Match points	Point-identification-tool. This is most important for the exact drawing of geometric figures.
Drawing Parameters...	Displays a dialog-box, which enables the user to set various parameters such as rotation-angle, homothety-modulus, number-of-graph-points, number-of-sides-of-regular-polygons, etc.
Drawing Sheet...	Displays a dialog-box, which enables the user to set various parameters such as sheet-dimensions, axes-origin-location, unit-lenght of axes-unis, labels-on-axes, etc.
Logo and Frame	Inserts two strings, into the rihgt and left bottom of the page, a separation line above the strings and a frame, defining margins.
Set Point-Mark...	Displays a dialog-box, enabling the user to set the figure (and frame/fill colors) of point-marks.
Delete All Point-marks	Deletes all Point-Marks (selected or not) of the document.


Match points command..(Settings menu)
Match Points  (Settings menu)

Use this command to force identification of nearby-points lying at a distance less than k cm.

Operation: 	Has two stages:

Activated: 	New points are forced to coincide with old lying nearby.

Deactivated	No effect.

Shortcuts

mouse	click the match-points-tool on the toolbar

Comments:

The most important tool (together with Set points) for precise drawing, since it forces coincidence of newly defined points with old existing ones.

The constant k, determining how far distant points should be identifyied, can be set through the appropriate field (measure in cm) in the Drawing Parameters dialog


Drawing parameters ... command..(Settings menu)
 Drawing parameters ...  (Settings menu)

Use this command to display a dialog-box, by means of which you can set the following constants:

Num of sides of regular polygons:	Number of sides of regular polygons. Default = 6.

Num of points on graphs:	Default = 50.

Left interval end (graphs):	Default = -5.0.

Right interval end (graphs):	Default = 5.0.

Translation-x(duplication):	How much is the duplicate translated. Default x = 0.5 cm.

Translation-y(duplication):	How much is the duplicate translated. Default y = -0.3 cm.

Center-x (rotation):	Center of rotation. Default x = 0.0.

Center-y (rotation):	Cetner of rotation. Default y = 0.0.

Angle of rotation:	Angle of rotation. Default = 30:00:00

Homothety center-x:	Center of Homothety. Default x = 0.0.

Homothety center-y: 	Center of Homothety. Default y = 0.0.

Homothety modulus:	Homothety-stretch-factor. Default = 1.5.

Identify points at distance <d:	Pts at distance less than d, are identified. Default d=0.3 cm.

Divide angle in N parts:	Default N = 2.

Ratio:	Produce two segments at ratio k. Default k = 0.5.

Comments:

You can any time visualize the rotation-center and homothety-center using the Set points tool, while holding down, respectively: ctrl+F1-key (for center of rotation) and ctrl+F2-key (for homothety-center).

You can avoid this dialog in defining the translation vector, using the data of a segment, as described in Duplicate .

You can avoid this dialog in defining the rotation-vector, using the data of an angle, as described in Rotate.

You can avoid this dialog in defining the homothety-data, using the data of a segment, as described in Homothety.


Drawing sheet ... command..(Settings menu)
 Drawing sheet ...  (Settings menu)

Use this command to display a dialog-box, by means of which you can set the following constants:

Sheet dimensions

Width:	Sheet-width in screen-pixels. Default = 770.

Height	Sheet-width in screen-pixels. Default = 1100.

Origin location

ox	location of axes origin, distance from left sheet-border in screen-pixels.

oy 	location of axes origin, distance from top sheet-border in screen-pixels.

Special origin location radio-buttons:

Middle of sheet 	(ox=width/2, oy=height/2).

Left-bottom 	(ox=0, oy=height).

Left-middle 	(ox=0, oy=height/2).

Bottom-middle 	(ox=width/2, oy=height).

Units on axis, in screen-pixels:

x-unit. Default = 50.

y-uint. Default = 50.

Number of labels per unit (integer or of the form 0.x, meaning 1 label every (1/0.x)-units).

x-unit. Default = 2.

y-unit. Default = 2.

Show-axes checkbox.

Show-labels checkbox.

Expansion settings, used when you press F11, to shrink, or F12, to extend the drawing sheet.

Expand both: 	Expand both the sheet and its units by the actuall homothety-stretch-factor.

Expand sheet only. 	Expand only the drawing sheet, leaving the same units.

Expand units only. 	Expand only the units, leaving the sheet dimensions invariant (default).

Button A4: Resets the sheet dimensions to the A4 standard (210 mm x 297 mm).


Logo and frame  command..(Settings menu)
Logo and frame  (Settings menu)

Use this command to insert the title and your logo in the bottom of the page.

The application inserts a line and two strings, in the following form:

__________________________________________________________________

actual document-title	Isoptikon ©, 1998

A frame defining margins around the sheet is inserted too.

Of course, you can modify or delete some of these objects, as you do with any other objects of the document.


Set point-mark ... command..(Settings menu)
 Set Point-Mark ...  (Settings menu)

Use this command to display a dialog-box, by means of which you can set the figure representing single points of the document (point-marks):

Select the figure:	3 Radio-buttons, select among three possible figures:

   Circle+cross (default)

   Cross,

   Circle.

Radius in cm:	Set the radius r of the circle representing the point: must  0.01<= r <= 1.0.

Thickness in pixels	Set the frame-thickness x : must be  0< x < 11.

Set actual frame/fill color: 	Check-box. If checked, the point-mark colors for frame and fill are set

to the actual general settings for shapes.


Delete all point-marks  command..(Settings menu)
Delete all point-marks  (Settings menu)

Use this command to delete all point marks (if any) you have previously set. This eliminates from the drawing all point-marks. It can be used at the end of an exact-drawing-session, where certain point-marks are usually set.


System menu  commands

The System menu contains the standard items featuring in every application running in the windows environment.

The menu is accessible by clicking on the small icon ...    on the top-left corner of the application’s window.

Restore	Restores the window to its previous size.
Move	Move the window.
Size	Size the window.
Minimize	Minimize the window.
Maximiize	Maximize the window.


System menu: Restore

Restore command (System menu)

Use this command to restore the window in its previous dimensions.


System menu: Size

Size command (System menu)

Use this command to display a four-headed arrow so you can size the active window with the arrow keys.

After the pointer changes to the four-headed arrow:

1.	Press one of the DIRECTION keys (left, right, up, or down arrow key) to move the pointer to the border you want to move.

2.	Press a DIRECTION key to move the border.

3.	Press ENTER when the window is the size you want.

Note:  This command is unavailable if you maximize the window.

Shortcut

Mouse:	Drag the size bars at the corners or edges of the window.


System menu: Move
Move command (Control menu)

Use this command to display a four-headed arrow so you can move the active window or dialog box with the arrow keys.

Note:  This command is unavailable if you maximize the window.

Shortcut

Key:	Press the Alt-key,

then the  LEFT-DIRECTION-key ,

the the letter M-key,

then

After the pointer changes to the four-headed arrow:

1.	Press one of the DIRECTION keys (left, right, up, or down arrow key) to move the pointer to the border you want to move.

2.	Press a DIRECTION key to move the border.

3.	Press ENTER when the window is the size you want.


System Minimize Command
Minimize command (application Control menu)

Use this command to reduce the application’s  window to an icon.

Shortcut

Mouse:	Click the minimize icon  [image]  on the title bar.


System Maximize Command
Maximize command (System menu)

Use this command to enlarge the active window to fill the available space.

Shortcut

Mouse:	Click the maximize icon  [image]  on the title bar; or double-click the title bar.


Switch to command (application Control menu)

Use this command to display a list of all open applications.  Use this "Task List" to switch to or close an application on the list.

Shortcut

Keys:	CTRL+ESC

Dialog Box Options

When you choose the Switch To command, you will be presented with a dialog box with the following options:

Task List

Select the application you want to switch to or close.

Switch To

Makes the selected application active.

End Task

Closes the selected application.

Cancel

Closes the Task List box.

Cascade

Arranges open applications so they overlap and you can see each title bar.  This option does not affect applications reduced to icons.

Tile

Arranges open applications into windows that do not overlap.  This option does not affect applications reduced to icons.

Arrange Icons

Arranges the icons of all minimized applications across the bottom of the screen.


Tangents Bisectrices  menu commands

The tangents-bisectrices  menu offers the following commands, relateted to the construction of tangents and bissectrices of isoptic cubics:

Bisectrices at	Tool to set a point P on an existing isoptic cubic x, and the bisectrices of x at P.
Tangent at	Tool to set a point P on an existing isoptic cubic x, and the tangent of x at P.
Tangents from	Tool to set a point P on an existing isoptic cubic x, and the tangents of x from P.


Bissectrices  command (Tangents menu)
Bissectrices at  command  (Tangents menu)

Use this command to set a point, say P, on an existing cubic, say x, and draw the bissectrices of x at P.

Operation: 	In one stage:

1st stage (draws): 	Click at the point A, near an isoptic-cubic x.

Comments:

The application finds the point P on x, next to the click-point A and draws the bissectrices of x at P.

Match points:

If pre-existing points nearby, readjustement of P.


Tangent at  command  (Tangents menu)

Use this command to set a point, say P, on an existing cubic, say x, and draw the tangent of x at P.

Operation: 	In one stage:

1st stage (draws): 	Click at the point A, near an isoptic-cubic x.

Comments:

The application finds the point P on x, next to the click-point A and draws the tangent of x at P.

Match points:

If pre-existing points nearby, readjustement of P.


Tangent from  command  (Tangents menu)

Use this command to set a point, say P, on an existing cubic, say x, and draw the tangents of x from P.

Operation: 	In one stage:

1st stage (draws): 	Click at the point A, near an isoptic-cubic x.

Comments:

The application finds the point P on x, next to the click-point A and draws the tangents of x from P.

Match points:

If pre-existing points nearby, readjustement of P.


Apollonian (isoptic) cubics

An application of group  theory to Euclidean geometry

By Paris Pamfilos (University of Crete) and

Apostolos Thoma (University of Ioannina)

1.Introduction

2.Defining equations

3.The cubic as an abelian group

4.Main properties

5.Singular isoptics

6.Reducible isoptics

7.Additional metric properties

8.Groups leaving the isoptic invariant

Bibliography on isoptic cubics


Introduction (Isoptics)

1.Introduction

1.0   The Apollonian circle c of two segments BA, AC on the same line e  figure-1, is the locus of points P of the plane for which the ratio |PB|/|PC| = |BA|/|AC|.

It is easily seen that for all P on c the line PA bissects the angle ang(BPC).  Thus the Apollonian circle is, at the same time, part of the locus of points P viewing the segments BA and AC under equal angles.

Considering more general positions for the segments AB, AC we arrive at the problem formulated  by Jacob Steiner  in 1852 :

1.1 Problem :  Given two line segments AB and CD on the same plane, find the locus of points P having the property : the angles viewing AB and CD from P, are equal (or supplementary) ang(APB) = ang(CPD) figure-2.

Apparently Steiner was not aware of the work of Van Rees  who, in 1829, studied, in another context, a family of cubic curves solving this problem.

In fact it is easily seen (Steiner) that the locus of points P with the above property is described by a pair of cubics passing  through the points A, B, C, D figure-3.

It was Gomes Teixeira in 1915 , who remarked that these cubics are precisely the curves studied by Van Rees in their normal (or canonical) form

1.2	     (x-a)(x^2 + y^2) + bx + cy = 0.

In fact,	using this canonical form, Van Rees proved several interesting properties of these cubics.

For the sake of completeness we give here all the proofs, using our method.

Brocard, Chasles, Dandelin, Darboux, Salmon and Teixeira  are among the people who studied these curves.

References to the older literature may be found in the Bibliography .

1.3   The main result on these cubics may be considered to be Theorem 4.6 (Van Rees) below.

A glance at the initial proof of this theorem, as well as the proofs of other properties, disperced in the above-mentioned literature, convinces of the lack of a unifying concept, relating algebraic to metric properties of these curves.

1.4   By our approach to the problem, which is fairly elementary, we try to show that the unifying concept, which allows a systematic discussion of the properties of these curves, is the group structure defined on a cubic, discovered by Jacobi  in 1835 (Fulton , Walker , Van der Waerden [8], Brieskorn, Knoerrer [9]).

This group structure and Bezout’s theorem are the only tools from algebraic geometry needed for our purpose.  They are briefly discussed in 3.

In 2. we derive the equations of the cubics, make some trivial remarks and give the pictures of the locus for some special positions of the points A, B, C and D.

In 4. we prove the main properties of the Apollonian cubics.

In 5. we discuss the cases in which one of the two cubics has a double point and in 6. we discuss the cases in which one of the cubics is reducible to a conic and a line.  The Apollonian circle of two segments, together with the supporting line of the segments, gives a special case of such a reducible cubic figure-1.

In 7. we discuss metric properties of the cubics and in 8. we examine a group generated by inversions, which leaves the Apollonian cubics invariant

.


Figure-1

The locus of points P of the plane

for which the ratio |PB|/|PC| = |BA|/|AC|


Figure-2

The locus of points P having the property :

the angles viewing AB and CD from P, are equal

(or supplementary) ang(APB) = ang(CPD)


Figure-3

The locus of points P having the property :

the angles viewing AB and CD from P, are equal

(or supplementary) ang(APB) = ang(CPD)

P is on one of these two cubics.


Bibliograph on isoptics
Bibliography on isoptics

[1] C. G. Jacobi,  De use Theoriae Integralium Ellipticorum et Integralium Abelionorum in Analysi Diophantea. Crelles Journal 13(1835) pp. 353-355.

[2] J. Steiner,  Aufgaben und Lehrsaetze. Gesammelte Abhandlungen, Bd II, p. 485. Chelse Publ. Comp. 1971.

[3] Van Rees,  Memoire sur les focales, Correspondance mathematique et physique (Quetelet)  V(1829), p. 361-378.

[4] G. Teixeira,  A propos de la question 1491,  Nouvelles Annales de Mathematiques, Serie III, 15(1915) p. 362.

[5] H. Brocard,  Nouvelles Ann. Mathem. 15( 1915) p. 138.

[6] W. Fulton, Algebraic Curves, W. A. Benjamin Inc. 1974.

[7] R. Walker,  Algebraic Curves, Springer Verlag, 1978.

[8] Van der Waerden,  Einfuehrung in die algebraische Geometrie, Springer 1973.

[9] E. Brieskorn, H. Knoerrer, Plane Algebraic Curves, Birkhaeuser 1986.


Defining equations (Isoptics)
2.The defining equations

2.1  We identify the plane with RxR‚ and represent points of the plane by pairs of real numbers A=(a1, a2), B=(b1, b2) etc.

We denote also by <X,Y> the usual inner product:

 <X,Y> = x1y1 + x2y2 .

2.2   The locus of points viewing the segment AB under the angle f, consists of two circles, desribed by the equations:

<X,X> - <X, (A+B)> + <A, B> = ±cotf(<X, J(B-A)> - <A, JB>).	 (*)

J denotes the +90-rotation of  which in coordinates is given by:

(x,y) |--> (-y, x).

The sign ± corresponds to the two sides of the segment AB (segments and circles are supposed to be oriented).

Analogous equations will describe the locus of points viewing CD under the angle f :

<X,X> - <X, (C+D)> + <C, D> = ±cotf(<X, J(D-C)> - <C, JD>).

2.3  Eliminating cotf among these equations we obtain the equations of the cubic curves in the form of determinants (intersections of pairs of circles: such cubics are called cyclic).

(<X,X>-<X,A+B>+<A,B>)(<X,J(D-C)>-<C,JD>)

- (<X,J(B-A)>-<A,JB>)(<X,X>-<X,C+D>+<C,D>)=0, 	(*)

(<X,X>-<X,A+B>+<A,B>)(<X,J(D-C)>-<C,JD>)

+(<X,J(B-A)>-<A,JB>)(<X,X>-<X,C+D>+<C,D>)=0. 	(**)

The two equations correspond to intersections of equally or inversely oriented circles viewing AB and CD under equal (or supplementary) angles.

2.4 Notice that the left columns of these determinants, equated to 0, represent, respectively,

the circles with diameter AB and CD :

<X,X> - <X, (A+B)> + <A, B>  = 0,

<X,X> - <X, (C+D)> + <C, D>  = 0,

whereas the right columns of the same equations, equated to 0, represent, respectively, the straight lines defined by AB and CD :

<X, J(B-A)> - <A, JB> = 0,

<X, J(D-C)> - <C, JD> = 0.

These remarks imply immediately the following:

2.5  Proposition  The two cubics (*) and (**), in 2.3,  solving the problem, have the points A, B, C and D in common.  Besides, if the circles with diameter AB and CD intersect, then their intersections are common to both cubics (points F, G in figure-4).  The same is true for the intersection point of the supporting lines of AB and CD (point E in figure-4).

2.6 Notice that the points of the cubic 2.3 (*) view also the segments AC and BD under equal angles.  In other words, this cubic is identical with the cubic which results by interchanging B and C.

In fact, a simple calculation shows that the coefficients of this cubic:

f(x,y) = C0 + C1x + C2y + C3x‚ + C4y‚  + C5xy + (x^2+ y^2)(C6x + C7 y) 	 (*)

are given by:

C0 =  	a2b1c1d1 - a1b2c1d1 - a1b1c2d1 -  a2b2c2d1 + a1b1c1d2 + a2b2c1d2 +

a2b1c2d2 - a1b2c2d2 ,	(0)

C1 =  - a2b1c1 + a1b2c1 + a1b1c2 + a2b2c2 -

a2b1d1 + a1b2d1 - a2c1d1 + b2c1d1 + a1c2d1 + b1c2d1 - a1b1d2 - a2b2d2 -

a1c1d2 - b1c1d2 - a2c2d2 + b2c2d2 , 	(1)

C2 = - a1b1c1 - a2b2c1 - a2b1c2 + a1b2c2 + a1b1d1 + a2b2d1 + a1c1d1 -

b1c1d1 + a2c2d1 + b2c2d1 - a2b1d2 + a1b2d2

 - a2c1d2 - b2c1d2 + a1c2d2 - b1c2d2 ,	(2)

C3 = a2b1 - a1b2 + a2c1 - b2c1 - a1c2 - b1c2 + a2d1 - b2d1 - c2d1+ a1d2 +

b1d2 + c1d2 ,	(3)

C4 =  a2b2 - a1b2 + a2c1 + b2c1 - a1c2 + b1ác2 -  a2d1 - b2d1 - c2d1 -

a1d2 + b1d2 + c1d2 ,	(4)

C5 =  2b1c1 - 2b2c2 - 2a1d1 + 2a2d2 ,	(5)

C6 = - a2 + b2 + c2 - d2,	(6)

C7 = a1 - b1 - c1 + d1 .	(7)

All these remain invariant (up to simultaneous sign change) under the permutation subgroup of the symmetric group S4 of the four letters A, B, C, D, generated by the permutations (expressed in cycles): (AB)(CD), (AC)(BD) and (BC).

Thus, the rearrangements of the letters in (2.3): (ACBD), (BADC), (BDAC), (CADB), (CDAB), (DCBA), (DBCA) leave this equation invariant.

This represents a subgroup of index 3 in S4 and has the following consequence:

Given four points A, B, C and D and taking the three possible pairs of segments (AB, CD), (AC, BD), (BC, AD) and solving the problem for each pair of segments, we find in total not 6 but only 3 different cubic curves.

This set of three different cubics, associated to the four points A, B, C and D, may be called Apollonian Configuration figure-5 of the four  given points.

2.7  Proposition   There are three different cubics belonging to the Apollonian configuration of the four points A, B, C and D.  These are given by (2.3(*)) and the equations which result from this one by rearranging the letters in the orderings (ABDC) (which gives (2.3(**)) and (BCDA).  Each of the three cubics is uniquely determined by the property of its points to view two pairs of segments under equal angles.

We call the three curves  CI, CII and CIII figure-5 , of the preceding proposition, Apollonian or isoptic cubics.  Each one describes the locus of points viewing two pairs of segments under equal angles.

The points of CI, for example, see AB and CD under equal angles and the segments AC and BD also under equal angles.

The only pair of segments (among the three possible with ends A,B,C and D) not viewed by CI under equal angles is (AD,BC), hence to specify this cubic we can use the symbol AP(AD,BC).

Simillarly  we use for CII and CIII, respectively, the symbols AP(AC,BD) and AP(AB,CD).

2.8   Applying proposition 2.5 to AP(AD,BC) = AP(BC,AD) we see that the intersection point H of AC and BD, as well as the intersection points I, J of the circles with diameters AC and BD, do belong to this cubic.

2.9   For special places of the segments AB and CD one gets interesting cubics as f.e. the singular (self-intersecting) isoptic cubic of figure-6 for which the defining segments AB and BC have a common end-point.

2.10  In general the Apollonian cubics can have two components or one. They can have double points, as in figure-6  and degenerate to a circle+line figure-7  or an equilateral hyperbola +"line at infinity" figure-8 .

2.11  The asymptotic line ux + vy + w = 0, of the cubic in (2.6)  is defined by the condition

lim(f(x,y) - (ux+vy+w)(x^2 + y^2))/(x^2 + y^2) = 0,    for (x,y) going to infinity.

Using (2.6) and taking limits we find

u=C6 , w=C7 and:

lim (C6x + C7 y) =

lim (C3x^2 + C4y^2 + C5xy )/ (x^2 + y^2) = (C3b^2  + C4a^2  - C5ba )/ (b^2 + a^2).

Thus, the asymptotic line will satisfy the equation :

C6x + C7 y + C8 = 0,  with

C8 = (C3C7^2  + C4C6^2  - C5C7C6 )/ (C7^2 + C6^2).

The asymptotic line, considered as a curve of the corresponding real projective space, is tangent to the cubic at the point (at infinity) [-C7, C6, 0]. Thus (anticipating Bezout’s theorem, discussed in 3) the cubic will have one more intersection point with this line, which will be real point K figure-9.

2.12   Notice that the oval of the cubic (if there is one) never meets the asymptotic figure-9.  In fact, in the contrary case the asymptotic line, counting multiplicities and the tangency at infinity, would have, at least, four points in common with the cubic, which by Bezout’s theorem is impossible.

2.13   An other important notion for cubic curves is that of flexes.  These are special points on a cubic c, which are defined as intersection points of c (expressed in homogeneous coordinates by a cubic homogeneous polynomial F(x0, x1, x2)=0), with the cubic defined by the determinant of the Hessian of c (given in homogeneous coordinates by):

G(x0,x1,x2) = det( (didjF), i,j=0,1,2 ).

(di, denoting partial derivative with respect to xi).

This is again a cubic curve and (anticipating again Bezout’s theorem) has 9 (real or imaginary) points in common with the original cubic (considered as a subset of the complex projective plane).

2.14   It is easily proved that the tangent at a flex has the multiplicity three, that for real cubics there are only one or three real flexes and that for every two flexes the intersection point of the line, joining them, with the cubic is a third flex (Van der Waerden,  p. 92) figure-10.

2.15  It can be proved that the curvature of the curve at a flex is zero.  This implies that the irreducible regular (without self-intersections) cubics, considered as subsets of the real projective plane, must have three real flexes (Brieskorn Knoerrer  p. 305).  There are however cases such that one of the flexes is at infinity (for Apollonian cubics, this implies that the cubic is symmetric with respect to some axis figure-11) .)


Figure-4

If the circles with diameter AB and CD intersect, then their intersections are common to both cubics (points F, G).  The same is true for the intersection point of the supporting lines of AB and CD (point E)


Figure-5

Isoptic configuration of four points A, B, C and D:


Figure-6

A singular isoptic, with self intersection at the common end-point of the AB, BC.


Figure-7

A degenerate isoptic, consisting of the union of the points of a circle and a line.

This happens f.e. when the segments AB and CD are not equal and lie on the same line.


Figure-8

A degenerate isoptic, consisting of the union of the points of an equilateral hyperbola

and the, so called line at infinity.This happens f.e. when the segments AB, CD are equal

and parallel. Equal viewing from points at infinity, means that for every fixed direction

(this thing corresponds to a point at infinity) the width between the parallels enclosing

AB and CD is the same


Figure-9

The asymptotic line of an isoptic cubic, meeting the curve (at most) at one point K.


Figure-10

The flex line of an isoptic cubic having (at most) three real flexes: E, F and G.


Figure-11

A symmetric about the line e isoptic cubic. Its asymptotic does not intersects the curve (K is at infinity) and there are at most two real flex points (the third is at infinity).


Abelian group structure (Isoptics)
3. The cubic as an abelian group

3.0 Bezout’s (1764) theorem is undoubtedly the cornerstone of the theory of plane curves.  This theorem relates the number of intersection points of two plane algebraic curves to their degrees.  In the theorem each intersection point is counted with its multiplicity, f.e. the intersection multiplicity of a tangent line to a curve at the point of tangency is two.

An extensive discussion of the concept of intersection multiplicity and a proof of this theorem may be found in [9], p. 227.

3.1  Bezout’s theorem  Let c and c' be two curves of the complex projective plane of degrees m and n respectively, which have no common component. Then they have mn intersection points (counted with multiplicities) .

3.2  A consequence of the theorem is that each straight line cuts (counting multiplicities) an irreducible cubic curve in exactly three points.  This leads to the idea to define an addition on the set of (regular) points of a cubic :

Given a cubic and taking a fixed point O on the curve, we can define an addition on the set of regular (i.e. points which have a uniquely defined tangent line) points of the cubic, which turns the (set of regular points of the) curve into an abelian group.

In fact, each pair of points (A, B) of the cubic defines a straight line which meets the curve in a third point C.  The line joining O and C meets the curve at a third point D which is defined to be the sum A+B (figure-12) .  It is proved that this operation is commutative, associative ([8] p. 90) and has O as its neutral element.  Jacobi [1] was the first who used this structure.

3.3  By choosing O to be a flex of this cubic and following this pattern  we see that the intermediate point C = -(A+B).  Thus, three points A, B and C of the cubic are on the same line if and only if figure-13

A + B + C = O.

3.4  In particular, an arbitrary point D of the cubic and the corresponding point E, at which the tangent of the curve at D meets again the curve, satisfy figure-14:

2 D + E = O,	(*)

and each one of the flexes of the cubic satisfies :

3 D = O.	(**)

A useful theorem generalizing the preceding formulas is the following ([8] p. 90):

3.5   Theorem  Take the neutral element O of the cubic c to be one of its flexes.  Then  3n regular points S1, ..., S3n  on the cubic, are the intersection points with a curve of degree n, if and only if

S1 +  ... + S3n = 0.

3.6  It should be stressed that the intersection points are considered to be points of the complex projective plane. Only in special cases all of them are real. Thus, a line, having one real intersection point with a cubic, has two additional intersection points in the complex projective plane.


Figure-12

Fix a point O on the cubic, and consider it as the zero of the abelian group. Each pair of points (A, B) of the cubic defines a straight line which meets the curve in a third point C.  The line joining O and C meets the curve at a third point D which is defined to be the sum A+B


Figure-13

The origin (zero) O of the cubic is a flex point, then A+B+C=O is equivalent with the fact, that A, B, C are the intersection-points of the cubic with a line.


Figure-14

The origin (zero) O of the cubic is a flex point, then 2D+E=O is equivalent with the fact, that a line e is tangent to the cubic at D and cuts it at E.  3F=O, characterizes a flex-point F on the cubic..


Main properties (Isoptics)
4. Main properties of Apollonian (isoptic) cubics

4.1   Theorem  The tangents at the points A, D of the isooptic cubic AP(AD,BC) meet on a point P(AD) of the cubic.  The tangents at the points B, C meet also on a point P(BC) of the cubic (figure-15).

Proof:   As noticed in 2., the curve AP(AD,BC) contains the intersection points E of the lines AB, CD and F of AC, BD (figure-15) :

A + B + E = O,  C + D + E = O,

A + C + F = O,  B + D + F = O.

Adding, substracting the equations and simplifying, we find that

2 A = 2 D  and  2 B = 2 C ,	 (*)

which mean that the tangents of the cubic at A, D meet on a point P(AD) of the cubic and the same happens also for the tangents at B and C which meet at  P(BC).

4.2   From (4.1) follows an other important property of AP(AD,BC), namely :

2(C - B) = O  and  2(D - A) = O.	 (*)

These mean that the points C - B and D - A have a tangent which passes through the zero point O.

Taking the O to be one of the flexes we see that there are at most three such different points say O1, O2, O3 (figure-16) .

Thus,

C = B + Oi  and  D = A + Oj  .	 (**)

From the equations in (4.1) we have also

A + C = B + D,

Thus, in (**) Oi = Oj and we proved :

4.3  Theorem  The four points A, B, C, D defining the cubic c = AP(AD,BC) satisfy the equations :

C = B + Oi  and  D = A + Oi  ,	       (*)

where Oi is one of the (at most) three points on the cubic, satisfying

2 Oi = O.

4.4   Actually the three points Oi lie on a line i.e. they satisfy:

O1 + O2 + O3 = O.

4.5   Concerning the ambiguity in the choice of Oi look at the remark 4.22 below.

4.6   We fix a flex O as the zero element of the group and denote by O1 the point given by (4.3).  For any point P on the cubic c, we denote by P* the point

P* = P + O1 .

This defines a transformation of the curve on itself which is involutive (equals its inverse):

(P*)* = P.

We call this the Main Involution of the cubic.

Thus, given A, B on the cubic AP(AD,BC), the other two points D=A* and C=B* are uniquely defined by means of the group structure on this curve.

4.7   Before to proceed to the next theorem we

must remark that every Apollonian cubic (considered as a curve of the complex projective plane) passes through the two cyclic points at infinity, represented in homogeneous coordinates by:

I =  [ 1, i, 0 ],

I' = [1, -i, 0 ].

In fact, substituting these for x, y and z in the homogeneous equation of the cubic, we see that they make it zero.

The same points lie also on every circle of the plane.  Thus,

4.7.1    Every circle passes through the two cyclic points at infinity.

4.7.2    Every Appolonian cubic passes also from the two cyclic points at infinity.

In general, a cubic passing throught these two points is called cyclic cubic. Thus, the Apollonian cubics are special cases of cyclic cubics.

Bezout’s theorem implies also that the intersection of a circle with an Apollonian cubic is a set of six points, two of them beiing I and I'. Let the other be U, V, W, Z. Then, after 3.5, they must satisfy (figure-17) :

U + V + W + Z + I + I'  = 0.

4.8  Theorem (Tangent-chord theorem)  Given the cubic c = AP(AA*,BB*), the angle between the chord AB and the tangent to c at A is equal (or supplementary) to the angle that the point A is viewing the segment A*B* (conjugate segment of AB) (figure-18) .

Proof:  Consider the circle which is tangent at A to the curve c and pass through B.  Let P be the other intersection point of the circle with c.  Then we have:

ang(XAB)= ang(APB) or ang(XAB) + ang(APB) = 180, and also

2A+B+P+I+I' = O. 	(*)

The point P lying on c has ang(APB)= ang(A*PB*) or ang(APB)+ ang(A*PB*) = 180.

From (*) we have also (since A+B = A*+B* (common intersection)):

A+A*+B*+P+I+I' = O.  	 (**)

Thus the points A, A*, B* and P lie also on some circle and so

 ang(A*PB* = ang(A*AB*) or ang(A*PB*)+ ang(A*AB*)=180.

We conclude that ÿXAB= ÿA*AB*.	q.e.d.

4.9   The following proposition gives us an important extra point H lying on the cubic AP(AD,BC).

4.10   Proposition  Let A, B, C and D four points in general position.  Consider the intersection points E of AB and CD and F of AC and BD.  The four circles passing, respectively, through the points {A,B,F}, {C,D,F}, {E,A,C} and {E,B,D} intersect at a point H (figure-19) ,  called Miquel point of the four circles  from which the segments AB, CD are viewed under equal angles and the segments AC and BD are also viewed under equal angles.

Proof: To prove this consider the intersection point H of the two circles (figure-19) {A,B,F} and {C,D,F}.  We have the equalities between angles :

f = ang(DFC) = ang(AFB)  = ang(AHB)  = ang(DHC), hence

ang(CHA) = ang(DHB).

We have also (quadrangles inscribed in circles) :

a = ang(FHB),  c = ang(DHF), hence

ang(DHB) = ang(DHF) + ang(FHB) = a +  c  = ang(CHA).

Thus, the angles ang(DHB) = ang(CHA) are supplementary to the angle ang(DEA) and consequently the circles {E,A,C} and {E,D,B} pass also through H.

From the preceding proposition follows that the point H belongs to the cubic c = AP(AD,BC).  We’ll see that H is an "important" point of this curve which is geometrically determined from A, B, C and D.

4.11  Proposition  The point H of the cubic c = AP(AD,BC) is one of the (at most three) tangent points of the tangents to c, from the point K at which the asymptotic line intersects the cubic (figure-20) .

Proof:  The proof is an easy consequence of theorem 3.5 . All Apollonian cubics intersect the line at infinity in the two cyclic points I, I'  and a third point, denoted by H*, and defined by the asymptotic direction.

In projective coordinates H* is described by [-C7, C6, 0] (relative to equation 2.6(*)).

Using (3.5) for the line at infinity, intersecting c at the points I, I' and H*, we have :

I + I' + H* = O.	 (*)

Using the preceding theorem for the circle {A,B,F} we have :

A + B + F + H + I + I' = O.	(**)

We have also

C + A + F = O      hence      A + F = -C

B - C + H - H* = O,

which by (4.3(*)) implies

H = H* + Oi    hence      2H = 2H* .

This proves our proposition.

4.12    We call H the center of the Apollonian cubic.  In the case of the degenerate reducible cubic of (figure-7) , we’ll see in 6. that H coincides with the center of the circle c.

4.13  Normal or Canonical form  It can be shown that taking the center H as origin of the coordinate system and the y-axis parallel to the asymptotic line, then the equation of the cubic AP(AD,BC) takes the form

 (x - a)(x^2+y^2)+bx+cy = 0.

4.14  It can be seen further that this is the equation of the cubic AP(AK,HH*), where A = K* and the "angle" under which a point P sees the infinite segment KH* is defined to be the angle between PK and the parallel to KH* (asymptotic direction) through P (figure-21) .

4.15  The following theorem (Van Rees 1829) may be considered to express the main property of the Apollonian cubics.

4.16  Theorem  Let A, B, C and D be four points on the plane, such that the cubic c = AP(AD,BC) is irreducible.  Let a flex O be the zero element of the group on c.  Then for any pair of points X, Y of c, the points W = Y* and V = X* on c are uniquely defined and the cubic AP(XV,YW) coincides with AP(AD,BC).

Proof:  Let P be the third intersection point of the line BX with the cubic c (figure-22) .  Then P+B+X = O implies (adding 2O1 = O) P + B* + X* = O.  Thus X*=V is the third intersection point of the line PB* with c.  Consider now the cubic AP(AD,XV ) (points viewing AX and DV under equal angles and also AV and XD ).  The following points of c are also points of AP(AD,XV):

A, X, A*, X*, -A-X, (-A-X)*, H, I and I' .

In fact, if Q is the third point of c on AX, then A+X+Q=O implies through 2O1=O, D+V+Q=O i.e. Q is the intersection point of AX and DV.  This shows that Q=-A-X is on the cubic.  Similarly if R = -A-X+O1= -A-O1-X = -D-X  and R is the intersection point of the segments XD and AV, also belonging to the cubic.

That H is on the cubic AP(AD,XV) is seen as follows :

Consider, as in the proof of 4.10, the circles {AXR} and {DYR}.  They intersect at a point H' belonging to AP(AD,XV).  We claim that H' = H.  In fact, let H" be the fourth intersection point of the circle {AXR} with c.

Then we have A+X+R+H"+I+I'=O and A+X*+R=O, which gives X-X * +H"-H*=O or H" = (H*)*=H.  Thus H is on {AXR} and analogously also on {DYR}, hence H" = H' = H.

The point P belongs also to AP(AD,XV).

In fact this follows immediately from the construction of P (figure-22) .  Thus, the two cubics c and AP(AD,XV) have 10 common points and by BezoutÕs theorem they are identical.  By the same argument we show that AP(AD,XV) is identical with AP(XV,YW).	q.e.d.

4.17   It is easy to show that when the cubic c=AP(AD,BC) has two components, then for every point P on c the angles viewing the two components from P are equal or supplementary (figure-23) .

[2A+P=O, 2B+P=O   hence  2A*+P=O, 2B*+P=O]

4.18   One can show also that the width d of the bounded component between parallel tangents to the asymptotic is equal to the corresponding width of the unbounded component (figure-24) .

4.19   It is easy to show that when the cubic c=AP(AD,BC) has two components, then for every point P of c the points P and P* lie in different components.  We conclude that H as well as O1 lie on the bounded component (if there is one).

4.20   Let c = AP(AD,BC) have one component. Let A be any point on c and let AB, AB* be the tangents from A to c, A*C, A*C* be the tangents from A* to c.

Prove that A, C and C* are collinear and BA*B* are collinear.

Prove that AC*C is the bissector (internal or external) of the angle ang(BAB*) and BA*B* is the bissector of the angle ang(CA*C*) (figure-25) .

[Hint :  2B+A=O Þ (adding 2O1=O) B+A*+B*=O.]

4.21  Theorem (Four tangents theorem)

Let c be an Apollonian cubic with two real components.  Let A be any point on the unbounded component of c.  Let AB, AB*, AC, AC* be the tangents from A to c.  Then BB* and CC* intersect at A* and BB* is orthogonal to CC* (figure-26) .

Proof : That BB* and CC* intersect at A* is easily seen (hint of preceding exercise).

Take an arbitrary point X on c (figure-26) .  Then ang(XA*B)= ang(X*A*B*) or ang(XA*B)+ ang(X*A*B*)=180.  But BA*B are collinear, so it follows that BB* is interior or exterior bissector of the angle ang(XA*X*).  Analogous is the proof for CC*.

4.22   Remark	The last theorem gives the answer to the problem of choice of the right point Oi.  In fact, when A=O (one flex of the cubic), then the four tangent points are O,O1,O2,O3 (figure-16) .  The theorem says that if Ok is the right point of order 2 then OiOj is orthogonal to OOk (i,j,k take the values 1, 2, 3) and this determines uniquely the Ok . In the above figure the right choice is k=1.  In the case c has just one component or it has a node there is just one point of order 2.

4.23   Let c=AP(AD,BC), then the points A, B, C and D lie on the same circle if and only if the tangent at the point of intersection of AB with CD is parallel to the asymptotic.

 [Hint : A, B, C, D on a circle is equivalent to  A+B+C+D+I+I'=O  which

is equivalent to  -E-E-H*=O (figure-27) ]


Figure-15

The tangents at A and D meet on a point on the cubic.

The tangents at B and C meet on a point on the cubic.


Figure-16

O is one of the flexes. There are (at most three) O1, O2, O3 points with 2Oi=O, i.e. their tangents pass through the origin. They lie on a line and one of the OOi is orthogonal to that line(Here the OO1).


Figure-17

The origin (zero) O of the cubic is a flex point, then 2D+E=O is equivalent with the fact, that a line e is tangent to the cubic at D and cuts it at E.  3F=O, characterizes a flex-point F on the cubic..


Figure-18

The angle between the chord AB and the tangent to c at A is equal (or supplementary) to the angle that the point A is viewing the segment A*B* (conjugate segment of AB):


Figure-19

f = ang(DFC) = ang(AFB)  = ang(AHB)  = ang(DHC)

ang(DHB) = ang(DHF) + ang(FHB) = a +  c  = ang(CHA).


Figure-20

The point H of the cubic c = AP(AD,BC) is one of the (at most three) tangent points of the tangents to c, from the point K at which the asymptotic line intersects the cubic:


Figure-21

A = K* and the "angle" under which a point P sees the infinite segment KH* is defined to be the angle between PK and the parallel to KH* (asymptotic direction) through P:


Figure-22


Figure-23


Figure-24


Figure-25

AC*C is the bissector (internal or external) of the angle ang(BAB*) and BA*B* is the bissector of the angle ang(CA*C*):


Figure-26

Then BB* and CC* intersect at A* and BB* is orthogonal to CC*:


Figure-27

A, B, C and D lie on the same circle if and only if the tangent at the point of intersection of AB with CD is parallel to the asymptotic:


Singular  (Isoptics)
5. Singular Apollonian (isoptic) cubics

5.1  Some singular cubics arise when B or C or D is identical with A.  If  A=D=(0,0) equation (2.6(*)) of the  cubic becomes :

(x^2+y^2)[(b2+c2)x-(b1+c1)y] - (b1c2+c1b2)(x^2-y^2) + (b1c1-c2b2)xy = 0.    (*)

The equation suggests to take the x-axis as bissector of the angle f formed by the segments AB, AC (figure-28) . Then we have for  0< f < pi :

b1 = r1cos(f/2),  b2 = -r1sin(f/2),  c1 = r2cos(f/2),  c2 = r2sin(f/2) . 	(**)

(x^2+y^2)[(r2-r1)sin(f/2)x- (r1+r2)cos(f/2)y] + 2r1r2xy = 0. 	(***)

5.2   Using this equation we can show immediately that the point A is singular, and that the coordinate axes are tangent to the curve at A.

Thus, the two tangents at the singular point are orthogonal.

5.3   Proposition  The irreducible Apollonian cubic  c  of two line segments AB and AC has a singular point at A and its tangents there are orthogonal and bissect the angle ang(BAC). (figure-28)   Besides c is also the Apollonian cubic of all pairs of segments AB', AC' whose endpoints are on c and their angle is bissected by the tangents at A.

5.4   The proof is a straightforward calculation, using either the general theory of 4. or the equation of the cubic in polar coordinates (r,t) which is :

  	r(t) =  r1r2sin(2t)/[r1sin(f/2+t)-r2sin(f/2-t)].

5.5  Using this proposition, we can find other segments, whose one of the Apollonian cubics is c.

In fact, with the previous notation, the points P of c, will view the segments BB' and CC' under equal angles (figure-29) .  This follows from the equalities of the angles at P :

ang(APB) + ang(APC) = pi, ang(APB') + ang(APC') = pi,   hence

ang(BPB') = ang(CPC').

It follows that the intersection points E, F of BB' , CC' and BC', CB', respectively, lie also on the cubic c. The following theorem shows that all singular (irreducible) Apollonian cubics are of the preceding kind.

5.6  Theorem  Let c = AP(BB*,CC*) be singular (figure-30) and let S be the singular point of c.  Then c = AP(SS,BB*).

Proof  :  We know that if c has a cusp at S then the group on c is isomorphic to R+(group with respect to addition) and if c has a node at S then the group on c is isomorphic to R*(group with respect to multiplication).

We know that B*=B+O1 and 2O1=O and so the group on c has an element of order 2.  Thus c can have only a node at S.  Now we make the following remark concerning the convergence of a point P on c to S:  If P->S then P*->S.  In fact, let f be the isomorphism of the group on c with R*.

Since f(O)=1, we have f(O1) = -1 (the only point of order 2 in R*).  But then f(P*) = f(P+O1) = f(P)f(O1) = -f(P).  The singular point S corresponds to 0 and the point at infinity.  But if f(P) ->0 then -f(P) ->0 and if f(P) ->infinity, then -f(P) ->infinity.  This proves our claim.

For any point Q on c we have ang(BQC) = ang(B*QC*) or ang(BQC)+ ang(B*QC*) =pi.  Assuming that C (hence also C*) converge to S, we get

 ang(BQS) = lim ang(BQC) = lim ang(B*QC*) = ang(B*QS) or analogously

ang(BQS) + ang(B*QS) =pi. C-> S , C* ->S.

This shows that c = AP(SS,BB*).

The singular irreducible cubics AP(AD,BC) can be also geometrically characterized :

5.7    Theorem  The irreducible cubic c = AP(AD,BC) is singular precisely when the segments AB, CD, AC, BD (figure-31) are tangent to the same circle.  In this case the center S of the circle coincides with the double point of the cubic and c = AP(SS,AA*).

Proof:  To prove this, consider first a singular isooptic cubic c, for two segments SD, SA, with a common end.  Let A, B, C, D four points such that c is isooptic for the segments AB and CD.  We know that

 ang(DCS) = ang(SCA), ang(DBS) = ang(SBA).

Thus, S is on the intersection of the bissectrices of the angles ang(DCA) and ang(DBA).  Similarly, ang(SAC) = pi - ang(SAB).  Thus SA is the exterior bissectrice of the angle ang(CAB). Analogously we show that SD is the exterior bissectrice of the angle ang(CDB).  Thus, all these bissectrices pass through S and the quadrangle ABCD admits an incircle with center S.

For the inverse, it is easy to see that if AB, CD, AC and BD are tangent to the same circle, then the center S of it will be a point of AP(AD,BC).  Now if we try to locate S* on AP(AD,BC) we will find S*=S, which can not happen if S is regular (in that case S* will belong to the group and we will have S=S*  hence  O1= O, a contradiction).  Thus we conclude that c will have a singularity (double point) at S.

5.8  Let c = AP(SS,BC) (figure-32) .  Show that BC intersects c at Y, so that SY is the altitude of the triangle BCS from S.  Show also that the median SM of this triangle is parallel to the asymptotic direction of c.


Figure-28


Figure-29


Figure-30


Figure-31


Figure-32


Reducible  (Isoptics)
6. Reducible Apollonian (isoptic) cubics

6.0   For special places of the points A, B, C and D we get reducible cubics AP(AD,BC) i.e. cubics whose equations decompose to the quadratic equation of a conic c and the equation of a line e.

Below we list all possible such decompositions.

The classification proceeds by considering the location of the cyclic points at infinity I and I', which belong to every Apollonian cubic. The details are easily supplied and are left as exercises.

6.1  When I, I' belong to the line e.  Then the line e coincides with the line at infinity, and one sees easily (using (2.6(*))) that the segments AC and BD have the same middle point and c is an equilateral hyperbola (figure-8) .  The hyperbola degenerates to two orthogonal lines, when the segments are equal and lie on e (figure-33) , or when the segments are opposite sides of a rombus (figure-34) .

6.2  When I, I' belong to the conic c then the conic is a circle and the line e goes through the center of the circle.

6.2.1  The four points A, B, C and D belong to the circle c and not to the line e.  Then we have the configuration of (figure-35) .  In this case e passes through the center of the circle and AB, CD are symmetric with rescpect to e.

6.2.2  The four points belong to the line and not to the circle.  Then we have the configuration of (figure-36) . In this case the circle has diameter OO', where O, O' are the points of e satisfying |OB|/|OC| = |OA|/|OD|.  This is a generalization of the classical Apollonian circle and gives an Apollonian circle for B=C.

6.2.3  The last possibility is that of (figure-37) .  In this case two points A, D belong to the circle c and the other two B, C belong to the line e.  Besides D and A are symmetric with respect to e.

6.3  Show that in the cases 6.2.1 and 6.2.3 the center of the cubic H coincides with the center of the circle c.  Show further that in case 6.2.3 there are two circles tangent to the sides AB, CD, AC, BD, whose centers are the intersection points of c with e (compare with theorem 5.7).


Figure-33


Figure-34


Figure-35


Figure-36


Figure-37


Additional metric properties  (Isoptics)
7. Additional metric properties

We start with the normal form of the cubics and certain results of Van Rees.

7.1  Proposition  For each secant through H,  meeting the cubic at the points P and P', the middle of PP' lies on a line e, which is parallel to the asymptote.  We call this the middle line of the cubic.  It lies in the middle between the asymptote and the parallel to it through the point H (figure-38) .

Proof:  Using the normal form

F(x,y) = (x-a)(x^2+y^2)+bx+cy = 0,	(*)

and taking the secant line in a parametric form  e(t) = tu , where u is a unit vector, we compute easily the roots of F(tu)=0 and the hafl-sum of them.  It turns out that

t+t'  =  a/u1   and   tt'  = (bu1+cu2)/u1,	(**)

from which follows that the line e meets the x-axis at x=a/2. 	q.e.d.

7.2   Corollary   For every point P of the Apollonian cubic, the line segment PP* has its middle on the middle line e.  The ordinates  x, x* of the points P(x,y), P*(x*,y*), refered to the axes of the normal form (7.1(*)), satisfy

x + x* = a.	(*)

Proof:  P+P'+H=O  implies  P*+P'+H*=O.  Thus P* is on the parallel to e through P'  (figure-38)

7.3   Corollary   The tangents parallel to the asymptote, tuch the cubic at points P, P* such that the line PP* passes through H.  In the case the cubic has two components, their "widths" parallel to the asymptote are equal (figure-39)

Proof:  Follows immediately from the previous corollary or the following:

P+P*+Q=O implies  2P+Q*=O, but 2P+H*=O since the tangent is parallel to the asymptote.  Thus, Q=H.

7.4   The property stated in 7.3 is characteristic for Apollonian cubics among the cyclic ones:

7.5   Theorem   Let c be a cyclic cubic with two real components, such that the width w of the bounded component between parallel tangents to the asymptote is equal to the corresponding width of the unbounded component. Then c is an Apollonian cubic (figure-39) .

Proof:  Let A, B, B* , A*,  be the four points such that their tangents are parallel to the asymptote and c' be the Apollonian cubic AP(AA*, BB*). In the intersection  of the two cubics belong the points of AB, A*B* and AB*, A*B (from the group structure of both cubics). But the Miquel point H of the points A, B, B*, A* belongs also to the intersection (propositions 4.10, 4.11 are true for cyclic cubics). Last, the real point at infinity of c belongs to the middle line of c' (see proposition 7.2 and the property of c stated in the Theorem). Thus, from proposition 7.2, belongs to the intersection. The two cubics c, c' have 10 points in common, hence they are identical (Bezout’s theorem).

7.6   Notice that this witdth can be measured in terms of a, b and c.  In fact, the points P , P* above are characterized by the equation

dF(P)/dy=0.	(*)

But the derivatives are

dF/dx = x^2+y^2+2x(x-a)+b,

dF/dy = 2y(x-a)+c.	(**)

Thus, dF/dy=0 implies y=-c/2(x-a) and introducing this into the equation (*) we find

4(x-a) ^2x^2+4bx(x-a)-c^2 = 0.	 (***)

This can be easily solved and gives the following values for x:

x =(asqrt(a^2+2(-bsqrt(b^2+c^2))/2.

Here everything depends on the expression

D = (a^2-2b)^2-4(b^2+c^2).	(****)

Assuming this to be non-negative, we find that the width is equal to

w =sqrt((a^2-2b-sqrt(D)) /2).

D controls the kind of the cubic i.e. wether it has one or two components or it is singular.  In fact, if D>0 then (7.6(***)) gives four real values for x and we have two components, each of width w.  This because the tangents parallel to the asymptote can’t be crossed again by the cubic.

7.7   If D=0 then (7.4(***)) gives three real values for x, namely (figure-40)

x =a/2    and   x= (a/2)±w	(*)

which is the case of a singular cubic.

Notice that in this case the middle line of the cubic coincides with the parallel to the asymptote passing through the singular point S of the cubic (the tangents at S -here the horizontal and vertical directions- are different from the x, y -directions of the normal form (figure-40)).

7.8   Finally, when w<0, (7.4(***)) gives two values for x and the curve has only one component (figure-41) .

7.9  Notice that in the case of two components the total width is

w’ = (a^2-2b)^2+sqrt(b^2+c^2),

whereas the "empty" width between the two components is

w’’ = (a^2-2b)^2-sqrt(b^2+c^2).

7.10   The constants a, b and c of the normal form are related also to the fundamental conjugation f(P)=P*.  In fact, the corresponding coordinates must both satisfy (7.1(*)),

which gives a linear system for the constants b, c.  This together with (7.3) gives the relations between the coordinates of P and P* (figure-42) :

xx*-yy*=b,

xy*+x*y=c.	(*)

7.11   Corollary  Let z and z* represent the points P, P* with complex numbers (with respect to the normal coordinate system of the cubic).  Then,

|z||z*| = b^2+c^2,

arg(zz*) = arctan(c/b) = v,	(*)

where v is the angle at K, between the asymptote and the line KH (figure-42).

7.12   Corollary   For all pairs of points P, P* the bissectrices of the angle ang(PHP*) are allways the same lines, namely the lines h and h' joining the touch-points of tangents parallel to the asymptotic line.

This may be considered as a special case of the four tangents theorem  and even a more general statement follows also directly from this theorem:

7.13   Theorem   For any points A, B on the Apollonian cubic c, the bissectrices of the angles ang(BAB*) are independent of the point B on the cubic and define two lines called l(A), l'(A) called bissectrices of A (figure-43) .

7.14  Notice that the middle line of the cubic can be considered as a bissectrice of the point at infinity H*.

7.15  Notice that in the case of two components, one cannot draw tangents from the bounded component.  In fact such a tangent would have four points in common with the cubic, and this is impossible.

7.16 Remark The first of equations (7.9) can be proved using proposition 4.10 (figure-44) .

In fact, the triangles AHB and DHC are similar and this implies |HA||HD|=|HB||HC|.

7.17  Solving (7.9) we find that P*(z*) can be expressed (in normal coordinates) as a Moebius function in z:

z* = f(z) = (b+ic)/(x+iy)	 (*)

or

z* = d/z,   where   d=b+ic.

This expresses the main involution as a Moebius transformation, which is defined everywhere on the plane (except H) and not only on the points of the curve.


Figure-38


Figure-39


Figure-40


Figure-41


Figure-42


Figure-43


Figure-44


Groups leaving invariant  (Isoptics)
8. Groups leaving the isoptic invariant

8.1  Proposition  Let c be an Apollonian cubic with two components and A, B be the points of tangency of the smallest circle containing the bounded component (figure-45) .  Then AB is a diameter of the circle and the point C, where AB intersects c has tangent parallel to the asymptotic line.  Furthermore let A', B' be the points where the tangents at A, B respectively, cut c.  A'B' is also parallel to the asymptote.

Proof: To prove this notice that  the smallest circle has either three contact points or two diametricaly opposed (why?).  Three is impossible (cubic) hence AB is a diameter.

A, B, I, I' belonging to the same circle    hence

2A+2B+I+I' = O	hence 	-2C-H*=O.

Also

2A+A'=O, 2B+B'=O    hence

A'+B' -2C = O        hence

A'+B'+H*=O.

8.2  Analogous is the situation for the greatest circle inscribed in the bounded component (figure-46) .  The tangent at the point F is parallel to the asymptote.  The tangents at the points D and E cut c, respectively, at points D' and E', such that D'E' is parallel to the asymptote.

8.3 Notice that 2A+A'=O implies 2A*+A'=O and since also A*+B*+C=O, this determines A*, B* by finding the tangent point of c from B' (figure-47) .  Another method to find the points A*, B* results from the tangent-chord theorem.  In fact, by this theorem, A and B will wiew the segment A*B* under a right angle, hence A, B, A* and B* lie on a circle and consequently AA* and BB* meet at the point G of c, where the tangent is parallel to the asymptote.

8.4  Notice that

A+G+B*=O       hence

A*+G*+B*=O  	hence

G*= C.

By corollary 7.3 we know that  the line CG goes through the point H. Since ABGH is on a circle, we have |CA||CB|=|CG||CH|= r‚, which may be considered to be "known", once c is given.  Thus the minimal circle v is orthogonal to a known circle with center C.  The same is true for the circle through ABB*A*.  The pairs of points (A,B), (A*,B*) and (G,H) are invariant with respect to the inversion (C, r‚) (figure-47) .

8.5   Proposition    The tangents to c at the points P where (C,r) cuts c (figure-48) pass through C.

Proof:   Notice that all secants through C, like AB, define A*B* such that the circle through ABB*C* is orthogonal to (C,r).  In fact |CA||CB|=|CG||CH| is still true.  Thus, taking A=B=P, we see that the tangent at the intersection point of c, with (C,r) passes through C.

8.6   Theorem  The involution (C,r) leaves the cubic c invariant.

In fact, the points AB are interchanged, for every secant CAB of c and the circle ABG goes allways also through H (figure-48) .

8.7  Corollary   The four points A, A*, B, B* where the tangents are parallel to the asymptote, together with the three points H, C and C*, where C is the third intersection point of the cubic with the line AB*, form an orthocentric configuration (figure-49)  i.e. B is the orthocenter of the triangle AA*B* and HCC* is the orthic triangle of AA*B*.

Proof:	The proof follows at once from the fact, that (A*, C*) and (A, C) are inverse points, with respect to the inversion leaving the cubic invariant and having center at B*. We’ll see that these seven points constitute the orbit of H under the group of eight elements to be defined below.

8.8  In fact, there are more inversions, related to minimal circles, leaving the cubic invariant.  These circles are shown in (figure-50) .  The centers of the inversions are the points A1, A2, A3, and A4 of the cubic, where the tangents are parallel to the asymptote.

BiCi are diameters of the corresponding minimal/maximal circles and the radii of the inversion circles satisfy the relations

ri‚ = |AiBi||AiCi|,         for i=1,...,4.	      (*)

8.9   Theorem  The inversions fi = (Ai, ri) defined above leave the cubic invariant and generate an abelian group G with 8 elements.  Except the generators and the identity, the other elements are the "translations of order 2 of the cubic",

f3of1= f1of3= f2of4= f4of2= f,

f2of1(P) = P+O3,

f3of4(P) = P+O3    and

f4of1(P) = P+O2,

f3of2(P) = P+O2.

Proof:  The compositions of the various inversions may be explicitly calculated.  In fact, for an arbitrary point P we have:

f1(P)+P+A1=O,

f3(f1(P))+f1(P)+A3=O,

2P+2A1-2f3(f1(P))-2A3=O,

2P=2f3(f1(P)).

Thus either f3of1=Id, which is impossible or f3of1(P) = P+Oi, for some i=1,2,3.  By taking P=A3 we have f1(P)=H and f3(H)=A1.  Thus, A1=A3+Oi,  which implies that i=1 and that the f3of1= f, the fundamental involution.

Obviously, we have also f3of1= f1of3.  The same argument shows that  f2of4= f4of2=f.

We have also

f1(P)+P+A1=O,

f2(f1(P))+f1(P)+A2=O,

2P+2A1-2f3(f1(P))-2A2=O,

2P=2f2(f1(P))

or f2of1(P) = P+Oi, for some i=1,2,3 and by a similar argument as before f2of1(P) = P+O3, f3of4(P) = P+O3.  Finally  f4of1(P) = P+O2, f3of2(P) = P+O2.

8.10 	In the case of a cubic with one component there are again four minimal circles but two tangents parallel to the asymptote (figure-51) .  Now the group G is generated by fi = (Ai, ri), i=1, 2 and a reasoning as above shows that f2of1(P) = P+O1 is the fundamental involution.

One can furthermore characterize the Apollonian cubics by means of their invariance, with respect to the two inversions defined above:

8.11   Theorem	  Let  c  be a real cubic that remains invariant  under two inversions and does not contain the line through the centers of the inversion.  Then  c  is an Apollonian cubic.

Proof:  Let  K1, K2 denote the centers and R1, R2 the radii of the two inversions I1, I2.  c has either one or three real points at infinity.  The image of the points at infinity under one inversion is the center of the inversion, so we conclude that K1, K2 will be points of  c.  Since cubic curves cannot have triple points, c has exactly one point at infinity H^, the points K1, K2 are simple points of c and the tangents at K1, K2 to c are parallel to the asymptotic line of c.

Let now A be any simple point of c, that does not belong to the two circles (Ki, Ri) and to the line through K1, K2.  Then, using the usual abelian group structure on the cubic,  we have for the points I1(A), I2(A), I1I2(A), I2I1(A) (figure-52) :

I1(A) + A + K1 = O

I1(A) = -A - K1

I2(A) + A + K2 = O

I2(A) = -A - K2

But

2K1 = 2K2 = -H^    hence

K2 = K1 + Oi,    where

2Oi = O    hence    Oi = - Oi  and

I2I1(A) + I1(A) + K2 = O

I2I1(A) - K1 - A + K2 = O

I2I1(A) = A + Oi .

Similarly,

I1I2(A) = A - Oi = A + Oi  Þ

I1I2(A) = I2I1(A).

Thus, the two circles are orthogonal  or

R1^2 + R2^2 = (K1K2) ^2 .

The line through K1, K2 intersects c at one further point, say H.  c remains invariant under I1, I2  and the line through K1, K2 remains invariant under I1, I2, so I1(K2) is one of the points K1, K2, H.  Since the two circles are orthogonal, the only choice is I1(K2)=H.  Similarly, I2(K1)=H.  Choose now A as before and consider the Apollonian cubic

c'=AP(AI1I2(A), I1(A)I2(A)).

It is easy to see that

K1, K2, H, I1(A), A, I1I2(A), I2(A)

belong to c', H is the center, I1I2(A)=A*, I2(A)=I1(A)* and K2=K1*.  If c and c' are different, then they intersect at 9 points, the above seven and two further points X1, X2.

Now, if we use the addition on c' we have

K1 + K2 + H + A + I1(A) + I2(A) +

I1I2(A) + X1 + X2 = O

- I - I' + X1 + X2 = O

H* + X1 + X2 = O.

So H* is on the line through X1, X2 and at the line at infinity.  Similarly, using the addition on c we have that

K1 + K2 + H + A + I1(A) + I2(A) +

I1I2(A) + X1 + X2 = O

and

K1+K2+H=O,

A+I1(A)=-K1,

I2(A)+ I1I2(A)=-K1,

-2K1=H^ ,

H^ + X1 + X2 = O.

So H^ is on the line through X1, X2 and on the line at infinity.

Thus, we conclude that H^=H*, or X1, X2 are the cyclic points I, I'.

In the first case, where H^=H*, we get  c=c', since K1, K2 have common tangents and should count two times as intersection points and so there are more than 9 points of intersection.

In the second case H^­H* and X1, X2 are the cyclic points I, I'.

Thus, c pass through the cyclic points.  Consider the cubic

c"= AP(K1K2, HH^).

Then, we can see that H is the center of it and

2K1+H^=O=2K2+H^ ,

which implies that c" is invariant also under the two inversions I1, I2 and I1I2(X)=X*, for all points of c".

To the intersection of c and c" belong the points K1, K2 (counted two times) H, H* = H^ , I, I' and at least one more point X, which should be real.  But since c and c" remain invariant under I1, I2, if X belongs to the intersection, then the same is true for I1(X), I2(X) and I1I2(X).  But the set

{X, I1(X), I2(X), I1I2(X)}

has always even number of elements except in the case that X is one of the points M1, M2 (figure-53) .  But if Mi is on c" then Mi is a singular point (since Mi=I1I2(Mi)=Mi*) so should count at least twice as intersection point, so we have that c"=c and c is always an Apollonian cubic.


Figure-45


Figure-46


Figure-47


Figure-48


Figure-49


Figure-50


Figure-51


Figure-52


Figure-53


Toolbar

The toolbar is displayed across the top of the application window, below the menu bar.

The toolbar provides quick mouse access to several tools used in Isoptikon,

To hide or display the Toolbar, choose Toolbar from the View menu.

Click	To

To get the Selection-tool.

Simple segment  -tool. Select this tool to draw segments.

Simple line  -tool. Used to draw lines. These are defined like the segments, but extend beyond the end-points of the segment, to the boundary of the drawing sheet.

General parallelogram  -tool, used to draw parallelogramms or rectangles.

General angle  -tool, used to draw angles.

General triangle -tool, used to draw triangles.

General circle -tool, used to draw circles.

Regular polygons  -tool, used to draw regular polygons.

Text-tool, used to define rectangular boxes containing text.

Segment rthogonal to  -tool. This tool allows to draw a segment orthogonal to an other existing segment or line.

Cross  -tool. Has the same function with the simple-segment-tool. Uppon release of the button, adds one orthogonal segment on the middle of the main segment. Used for work with the middle of a segment.

Tangent to circle -tool. Selects a circle and draws its tangents from a given point.

Tangent to circle/line _pass 2 pts -tool. Selects a circle/line/segment and draws circles tangent to the selected object and passing through two other points.

Tangent to circle/line centered at -tool. Selects a circle/line/segment and draws circle(s) tangent to the selected object and centered at a given point.

Enclosing circle -tool. Selects an existing triangle and draws its enclosing circle.

Divide angle -tool. Selects an existing angle or polygon-angle (exterior-interior) and divides it in k (default=2) parts.

Reflexion -tool. Use this command to select a line/segment/polygon-side, say x, then, select some objects and reflect them on x (actually on its supporting line).

Inversion -tool. Use this command to select a circle, say x, then, select some objects (actually only circles, lines and segments are affected by this transformation) and invert them on x.

Match-points -tool. This is most important for the exact drawing of geometric figures. It allows precise coinsidence of points.

Send to back-tool. Select first some object (with the selection tool) and press this button to send it behind all other objects.

This tool is not through a menu-item available:

Send to front-tool. Select first some object (with the selection tool) and press this button to send it above all other objects. This is usefull, when picking overlapping objects. It is allways the object lying above all selected.

This tool is not through a menu-item available:

Intersection -tool. Select first two objects (selection-tool) and then press this button to find out their intersection points.

Duplicate -tool. Select first some objects (selection-tool) and then press this button to duplicate them (produce exact copies).

Or, select a segment and press this button, while holding down the shift-key, to set the translation vector equal to the selected segment. This method avoids the use of Drawing Parameters... dialog.

Rotate -tool. Select first some objects (selection-tool) and then press this button to rotate them.

Or, select an angle and press this button, while holding down the shift-key, to set the rotation-center and rotation-angle, to the data of this angle. This method avoids the use of Drawing Parameters... dialog.

Homothety -tool. Select first some objects (selection-tool) and then press this button to their homothetical transforms.

Or, select a segment and press this button, while holding down the shift-key, to set the homothety center equal to the first end-point of the selected segment, and the homothety-modulus equal to the length of the selected segment. This method avoids the use of Drawing Parameters... dialog.


Triangles  menu commands

The triangle menu offers the following commands, relateted to the construction of triangles:

General	Tool to draw a triangle by defining its base AB and the corner C, lying opposite to AB.
Through a,b,c...	Displays a dialog, by means of which the user can define a triangle through three of its principal elements (sides or angles).
Medians of	Tool to draw the medians of a previously defined triangle. The tool acts as a selection tool.
Heights of	Tool to draw the heights of a previously defined triangle. The tool acts as a selection tool.
Bisectrices of	Tool to draw the bisectrices of a previously defined triangle. The tool acts as a selection tool.
Euler circle	Tool to draw the Euler circle of a previously defined triangle. The tool acts as a selection tool.
Euler line	Tool to draw the Euler line of a previously defined triangle. The tool acts as a selection tool.
Inscribed circles	Tool to draw the four inscribed circles of a previously defined triangle. The tool acts as a selection tool.
Enclosing circle	Tool to draw the enclosing circle of a previously defined triangle. The tool acts as a selection tool.


General triangle  command (Triangles menu)
General  command (Triangles menu)

Use this command to create a new triangle, with base, say x=AB, and opposite to x vertex, say C.

Operation: 	In two stages:

1st stage (draws): 	Click somewhere, say A,...drag...release, say at B.

2nd stage (draws): 	Click somewhere, say B1,...drag...release, say at C.

A is the first end. B is the second end of the base (segment)  x=AB.

C is the vertex of the triangle, opposite to the side x, constructed in the first stage.

Shortcuts

mouse	click the triangle-tool on the toolbar

Comments:

The first stage produces a segment. All rules for segment-construction apply here.

In the second stage, holding down the cttl-key, produces rectangular (at C) triangles.

In the second stage, holding down the shift-key, produces isoscelii triangles (having CA = CB).

Match-points enabled:

If pre-existing points nearby, readjustement of all three A, B and C.


Through a, b, c  command (Triangles menu)

Use this command to display a dialog-box, where you can define a triangle by setting 3 of its principal elements (angles or sides).

The triangle is supposed to be labeled by ABC, with vertices A, B, C and sides a=BC, b=CA, c=AB.

The dialog contains 5 radio-buttons, for selecting the triangle-elements by means of which you want to construct the triangle:

1)  Three sides a, b, c (case SSS)

2)  Two sides and an angle a, b, C (case SAS)

3)  Two angles and a side B, C and a (case ASA)

4)  Two sides and an angle (ambiguous case, the angle is not the contained by the two sides) A, b and a.

5)  Three angles (ambiguous: the triangle is modulo similarity defined) A, B, C.

The dialog contains also a check-box: Angles in degrees.

If this is checked, then angles appear in degrees, otherwise they appear in radians.

There are three fields, where you enter data.

The interpretation of these data depend on your choice (radio-buttons and check-box above).

For your convenience, on the left of these fields you see indications on what they expect.

If the data you enter are invalid, nothing is drawn.

There is a button : Old-settings,

by which you return to the old settings of the dialog.


Medians of  command (Triangles menu)
Medians of  command (Triangle menu)

Use this command to construct the medians of an existing triangle.

Operation: 	In one stage:

1st stage (selects): 	Click somewhere to select a triangle, say x.

The application inserts the three medians of x.


Heights of  command (Triangles menu)
 Heights of  command (Segments menu)

Use this command to construct the heights of an existing triangle.

Operation: 	In one stage:

1st stage (selects): 	Click somewhere to select a triangle, say x.

The application inserts the three heights of x.


Bissectrices of  command (Triangles menu)
Bisectrices of  (Triangles menu)

Use this command to construct the bissectrices of an existing triangle.

Operation: 	In one stage:

1st stage (selects): 	Click somewhere to select a triangle, say x.

The application inserts the three inner-bissectrices of x.

Comments:

Alternatively, you can use the Divide angle tool to divide individual angles (internal or external of a triangle).


Euler circle  command (Triangles menu)

Use this command to construct the Euler circle of an existing triangle.

Operation: 	In one stage:

1st stage (selects): 	Click somewhere to select a triangle, say x.

Comments:

The application inserts the Euler circle of x.


Euler line  command (Triangles menu)

Use this command to construct the Euler line of an existing triangle.

Operation: 	In one stage:

1st stage (selects): 	Click somewhere to select a triangle, say x.

Comments:

The application inserts the Euler line of x.


Inscribed circles  command (Triangles menu)

Use this command to construct the four inscribed circles of an existing triangle x.

Operation: 	In one stage:

1st stage (selects): 	Click somewhere to select a triangle, say x.

Comments:

The application inserts the four inscribed circles of x.


Enclosing circle  command (Triangles menu)
Enclosing circle  (Triangles menu)

Use this command to construct the enclosing circle of an existing triangle, say x.

Operation: 	In one stage:

1st stage (selects): 	Click somewhere to select a triangle, say x.

Shortcuts

mouse	click the enclosing-tool on the toolbar

Comments:

The application inserts the enclosing circle of x.


View menu commands

The View menu offers the following commands:

Toolbar	Shows or hides the toolbar.
Status Bar	Shows or hides the status bar.
Line thickness	Shows or hides the line-thickness toolbar.
Color tools	Shows or hides the color toolbar.
Font tools	Shows or hides the font toolbar.


View Toolbar command
Toolbar command (View menu)

Use this command to display and hide the Toolbar, which includes buttons for selecting most frequently tools, used in the application, such as the selection tool,line-tool,triangle-tool etc.  A check mark appears next to the menu item when the Toolbar is displayed.

See Toolbar for help on using the toolbar.


View Status Bar Command
Status Bar  command (View menu)

Use this command to display and hide the Status Bar, which describes the action to be executed by the selected menu item or depressed toolbar button, and keyboard latch state.

A check mark appears next to the menu item when the Status Bar is displayed.

See Status Bar for help on using the status bar.


Status Bar

The status bar is displayed at the bottom of the application’s window.

To display or hide the status bar, use the Status Bar command in the View menu.

The left area of the status bar describes actions of menu items as you use the arrow keys to navigate through menus.

This area similarly shows messages that describe the actions of toolbar buttons as you depress them, before releasing them.

If after viewing the description of the toolbar button command you wish not to execute the command, then release the mouse button while the pointer is off the toolbar button.

The right area of the status bar indicates the coordinates of the mouse, with respect to the axes of the drawing-sheet.


Line thickness
Line toolbar  command (View menu)

Use this command to display and hide the Line-toolbar, which determines the pen-width in pixels (default = 2).

A check mark appears next to the menu item when the Status Bar is displayed.

The Line-thickness-bar has the form:

The buttons select 1, 2, 3, 4, 5 and 6 -pixels-wide pens.

The  ?-button displays a dialog, where you can set an arbitrary thickness.

The other buttons set the pattern for dashed lines.

These patterns are currently only for 1-pixel-wide pens possible.


Color toolbar
Color toolbar  command (View menu)

Use this command to display and hide the Color-toolbar, which determines the pen-color, for frames of shapes, the fill-color, the text-color and the axes color.

A check mark appears next to the menu item when the Color-toolbar is displayed.

The Color-toolbar has the form:

The buttons select the colors (!).

We apologize for the inclusion of Byron’s verses (instead of explanation what the red-, white, or brown- colors are ).

The No-button, means no color (transparent).

The four last buttons determine which objects are affected by a color change, in that order:

Frame-color,

Fill-color,

Text-color,

Axes-color.


Font toolbar
Font toolbar  command (View menu)

Use this command to display and hide the Font-toolbar, which determines the font, size, style and alignement for text-boxes.

A check mark appears next to the menu item when the Font-toolbar is displayed.

The Font-toolbar has the form:

The first four buttons define the style:

Bold,

Italics,

Underline,

Strike-through.

The next three define the alignement in text-boxes:

Left justified,

Centered,

Right-justified.

The wide combo-box defines the current font.

The last combo-box defines the letter size of combo-boxes.