Invent geometry tasks that require a proof

The properties detected with OK Geometry are a rich source of proving tasks in planar geometry. 

Observing even a simple configuration with OK Geometry brings to light many of its properties. Some of them can be trivial or, for some reason, irrelevant, but we can often find among them interesting, nontrivial and perhaps unexpected properties that can serve as proving task.

Here is an example. Consider the following configuration:

Let O be the circumcentre of a given triangle ABC and let D be the intersection of lines CD and AB. Finally, let E and F be the orthogonal projections of D ont sidelines AC and BC respectively.


The observation of this configuration with OK Geometry produces a long list of properties. We selected some of them and shape them as proving tasks:


 
Prove that the triangles ADE and DFC are similar.

 

 
Prove that the line EF os parallel to the baseline AB.
 


 
Prove that the area of triangle ADC is twice the area of triangle EOC.

 


Prove that the points D, F, C, E lay on the same circle.

 


Prove that the line DE is tangent to the circle through points B, C, D.


Prove that the circle through A,E,F, the circle through D,O,F, and the line through A,O meet in a common point.