# Circle, Isosceles Triangle and a Fixed Point

What is this about?

A Mathematical Droodle

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Copyright © 1996-2018 Alexander BogomolnyThe 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

For details see a wiki write-up. There are also five extra solutions.

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Copyright © 1996-2018 Alexander Bogomolny71625270