LocusEquation[AreCongruent[α,β],C] determines where to put C in order to the angles α and β are equal. With no doubt C must take place on the bisector of AB. In such cases the triangle will be isosceles.
There is, however, one remarkable issue. GeoGebra draws not only the bisector of AB, but also AB, as the final result. The reason is that instead of computing α=β, cos(α)=cos(β) is used. This problem has its roots in complex algebraic geometry, and currently cannot be handled elegantly in GeoGebra.