Observing formulae in dynamic constructions

OK Geometry now observes complex algebraic relationships between geometric quantities in a dynamic construction.

Observe formulae, a new feature introduced in the version 18.1.10 of OK Geometry, observes complex algebraic relationships between selected geometric quantities in a dynamic construction. Observe formulae command comes in three variants that differ only in the way the geometric quantities are conveyed.

• The simple variant searches for algebraic relations involving specified quantities, which were beforehand measured in the Sketch Editor.
• The advanced variant looks for algebraic relations that involve specified geometric quantities that need not to be measured beforehand.
• The triangle variant searches for algebraic relations that relate a given geometric quantity or the ratio of two given quantities of a triangle to various unspecified geometric quantities of this triangle.

Here is an illustration of the use of the Observe formulae command.

Given is a triangle ABC. Let ri be the radius of its incircle. Furthermore, let ra be the radius of the circle that touches the sides AB, AC and the incircle and lays between A and the incircle. Define rb and rc cyclically. We wonder if there is a nice relationship between the radii ra,rb,rc,ri.

The simple variant of the command Observe formulae requires that in the dynamic construction we measure the radii of the four circles. Observe formulae then looks for algebraic relationships between the four radii and detects that:
 

ri=√(ra∙rb)+√(rb∙rc)+√(rc∙ra)
 

The advanced variant of the command Observe formulae performs a similar task, except that the considered quantities do not have to be measured beforehand in the Sketch editor.

The triangle variant detects, among others, the following relationships (rewritten after an algebraic manipulation):
 

ra=r∙sin⁡(A/2)∙sin⁡(B/2)∙sin(C/2)∙(1-sin(⁡A/2))/(1+sin(⁡A/2))

ra=ri∙(1-sin⁡(A/2))/(1+sin⁡(A/2))
 

where r is the radius of the circumcircle of the triangle ABC.
 

Observe formulae merely observes whether geometric quantities satisfy some relations that can be (in some way) reduced to polynomial expressions. It does not prove the relations and it may also overlook existing relations. Thus, it is a useful heuristic tool for obtaining hypotheses, that should be proved in a mathematically decent way.

More on Observe formulae command can be fount here and in the OK Geometry Plus Reference Manual.