Converse of Thales' circle theorem and the theorem of the inscribed angle in a triangle

Classic introduction of Thales' circle theorem

Problem 708 from Austrian textbook [i]mathematiX[/i] (Boxhofer-Huber-Lischka-Panhuber, Veritas 2013) for pupils at age 12

A possible introduction of the converse with GeoGebra (joint work with Katharina Schiffler)

Use the Point tool [icon]https://www.geogebra.org/images/ggb/toolbar/mode_point.png[/icon] to check if the third vertex [i]C[/i] of triangle [i]ABC[/i] makes the triangle right.

Generalization

An angle [i]α[/i] is given. Points [i]C[/i] in the plane to be found such that the angle [i]ACB[/i] equals to [i]α[/i].

If α=90°, the special case [i]Converse of Thales' circle theorem[/i] will be observed.[br][br]The command [code]LocusEquation[AreCongruent[α, β], C][/code] has been used in GeoGebra here. In the background an algebraic equation system must be solved, thus there is no difference here between equations cos[i]α[/i]=cos[i]β[/i] and [i]α[/i]=[i]β[/i]. For this reason there will be two circles shown in general.

Proof with the Relation tool

Information: Converse of Thales' circle theorem and the theorem of the inscribed angle in a triangle