Draw 'difficult' objects




Create a circle tangento to three other circles, the radical axis of two circles, the Apollonius circle...


OK Geometry is designed for analysing dynamic geometric configurations. Besides standard construction commands (e.g. perpendiculars, intersection of objects), OK Geometry contains three mechanisms for creating difficult objects, i.e. objects related to non-straightforward constructions  (e.g. a circle tangent to 3 given circles). With these tools one can easily obtain a configuration containing difficult objects and then study it. Here are the three metods:


Objects defined by 'touching' objects

One way to create lines and circles is via 'tuching' objects. The term 'touch' comprises a veriety of relations: touching a point (=passing through a point), touching a line (=parallel to a line, tangent to a line, perpendicular to line), touching a circle (=tangent to a circle, radical axis). The same command  (button) thus gives rise to several constructed objects - to distinguish between them ure repeatedly the Alt button or mouse-scroll.
Here are some illustrative examples:

The command Circle 3 objects (button ?) positions a circle so that it touches 3 given objects (point, lines, circles). Touching 3 points results in a circumcircle of a triangle, touching 3 lines results in an inscribed or exscribed circle. In the figure above the created circle 'touches' the point P, the circle  and the line p.

 

Use Alt button or mouse-scroll to choose among possible solutions.


Line 2 objects (button ?) positions a line so that passes through given points, is tangent to given circles or is parallel or perpendicular to a given line. In the figure above the created line 'touches' the line p and the circle k,

 

Use Alt button or mouse-scroll to choose among possible solutions.


If the command Line 2 objects (button ?) is applied to 2 circles you can obtain common tangents as well as the radical axis of the two circles. 

 

Use Alt button or mouse-scroll to choose among possible solutions.

 

Often used constructions

In planar geometry we use various kinds of transformations (e.g. translations, mirroring, similitudes, projectivities) and basic operations (e.g. angle bisector, harmonic conjugate). Several often used constructions are readily available in OK Geometry, so that you can obtain in a few steps a configuration to be observed and studied.  Some examples of operations are shown below.


For the given points A, P, B, we used the command Apollonius circle to obtain the set of point T, for which |AT| : |BT| = [AP| : |BP|.

The command Conic points shows the characteristic points of conics (foci, centre, etc.). In the example above, the trisectors (green points) of the sides of tringle ABC lay on a conic. We obtained its characteristic points. The triangle observation shows that the centre of the ellipse is the centroid of ABC and the main axis of the ellipse is the Steiner axis of the triangle.  

Among various kinds of available transformations of the plane there is also the inversion in circle. In the example above, the line p is inverted in circle k, the resulting object is the circle p'.

Standard shapes

Several shapes that are often used in geometric constructions are readily available in OK Geometry (command Advanced|Shapes|...). Below are three examples of available shapes. Whether they are based on more or less trivial constructions, using them can save considerable construction time.


To draw an equilateral triangle, a command in the group of commands Advanced|Shapes is used. The third vertex and the triangle itself is obtained from two vertices.

To draw the shape Bicentric quadrilateral in the group of commands Advanced|Shapes, the approximate positions of the four vertices have to be set. Two of the vertices are then modified to obtain a bicentric quadrilateral. The inscribed and circumscribed circles are not part of the construction. 

There are several ways to create and ellipse (and a conic in general). The common way is to provide 5 points on the conic. The example above shows an ellipse defined by 5 tangent lines (three lines passing through the trisectors of sides of a triangle and 2 sidelines of the triangle).

Example 1. Mixtilinear incircle

Given is a triangle ABC. The A-mixtilinear incircle  is the circle internally tangent to the circumcircle and to sidelines AB and AC.

Mixtilinear incircles can be readily obtained with the command Circle 3 objects. Given a triangle ABC, the the command can be used to obtain the circumcircle of ABC (since it 'touches' the points A, B, C). The same command gives rise to A-mixtilinear incircle (since it touches two sidelines and the circumcircle, eventually followed by Alt  command or mouse-scroll to choose among the several circles that touche the two sidelines and the circumcircle.

The configuration can be observed to obtain hypotheses of the points of contact of the mixtilinear incircle with the circumcircle and the sides of triangle or the centre of the mixtilinear incircle.
 

 

We present the observation of the point P of contact of the A-mixtlinear incircle with the circumcircle of ABC. See Advanced construction hints to learn how to proceed.

Here is a useful observation. The related construction of point P is self evident. Obviously, the observed property has to be proved (syntetically or with an automated prover). 



Example 2. An advanced triangle problem


Given is a triangle ABC and its inscribed circle. Consider the three (small) circles, each tangent to two of the triangle's sides and externally tangent to the incircle. Finally, consider the (big) circle, internally tangent to the three (small) circles.

For the (small) circle tangent to AB and AC, let A' and A'' be its points of contact with the incircle and the (big) circle. Define B', B'' and C',C'' cyclically.

The configuration is easily worked out with OK Geometry, so we can explore it, in particular the centre P of the (big) circle.   



We present some results of the Advaned triangle observation (the method is presented elsewhere (see here)). We use the notation of the triangle centres from the ETC (Enyclopedia of triangle centres). The results are also expressed in terms of various triangle operations, whicha are available in OK Geometry and explained in its Glossary.
  1. The lines AA', BB', CC' obviously concur in the incentre of the triangle ABC, which is the centre X1 in ETC.

  2. he lines AA'', BB'', CC'' concur in a point Q, which is a known centre of triangle ABC. In fact, Q is the centre X8241 in ETC (the homothetic centre of Hutson-intouch triangle and the tangential-midarc triangle). The point Q is collinear with the centres X1 and X167.

  3. The lines AA'', BB'', CC'' concur in a point R. It is not (yet) one of the ETC centres of ABC. OK Geometry claims that R is the intersection of several lines through known centres, e.g.  (X1,X167) and (X1699,X8149). R is also the image of the centre X7022 under the projectivity ABCX7 › ABCX1.

  4. The point P is also not (yet) and ETC centre. OK Geometry claims that P is the intersection of the following two lines: the first one is the line through the RTC centres X1 anf X167 (i.e. the line through Q and R); the second line contains the Ceva conjugate of Q and X1 and the Ceva conjugate of R and X1.


The obtained information path a way to the Euclidean construction of the studied configuration and also allow the computation of the trilinear coordinates of the points Q, R and P.

Download figures (png)

Download constructions (p)