Circle, Isosceles Triangle and a Fixed Point
What is this about?
A Mathematical Droodle
22 October 2016, Created with GeoGebra
The applet may suggest the following a problem from the 2006 Irish MO:
P and Q are points on the equal sides AB and AC respectively of an isosceles triangle ABC such that AP = CQ. Moreover, neither P nor Q is a vertex of ABC. Prove that the circumcircle of the triangle APQ passes through the circumcenter of the triangle ABC.
Let C(APQ) be the circumcircle of Δ. Let O be the second intersection of C(APQ) with the altitude from the apex A. Consider two triangles: BPO and AQO.
The idea is to prove that the triangles are congruent, from which it would follow that