Converse of Thales' circle theorem and the theorem of the inscribed angle in a triangle
Classic introduction of Thales' circle theorem
A possible introduction of the converse with GeoGebra (joint work with Katharina Schiffler)
Use the Point tool 
to check if the third vertex C of triangle ABC makes the triangle right.
to check if the third vertex C of triangle ABC makes the triangle right.Generalization
An angle α is given. Points C in the plane to be found such that the angle ACB equals to α.
If α=90°, the special case Converse of Thales' circle theorem will be observed.
The command
LocusEquation[AreCongruent[α, β], C] has been used in GeoGebra here. In the background an algebraic equation system must be solved, thus there is no difference here between equations cosα=cosβ and α=β. For this reason there will be two circles shown in general.