IMO 2009 Problem 2

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 $OP = OQ.$

Proposed by Sergei Berlov, Russia