2009 IMO Problems/Problem 2

Problem

Let $ABC$ be a triangle with circumcentre $O$ . The points $P$ and $Q$ are interior points of the sides $CA$ and $AB$ respectively. Let $K,L$ and $M$ be the midpoints of the segments $BP,CQ$ and $PQ$ , respectively, and let $\Gamma$ be the circle passing through $K,L$ and $M$ . Suppose that the line $PQ$ is tangent to the circle $\Gamma$ . Prove that $OP=OQ$ .

Author: Sergei Berlov, Russia

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2009 IMO Problems/Problem 2

Problem