CN-TST 2011 Problem 16

Let $H$ be the orthocenter of an acute trangle $ABC$ with circumcircle $\Gamma$ . Let $P$ be a point on the arc $BC$ (not containing $A$ ) of $\Gamma$ , and let $M$ be a point on the arc $CA$ (not containing $B$ ) of $\Gamma$ such that $H$ lies on the segment $PM$ . Let $K$ be another point on $\Gamma$ such that $KM$ is parallel to the Simson line of $P$ with respect to triangle $ABC$ . Let $Q$ be another point on $\Gamma$ such that $PQ \parallel BC$ . Segments $BC$ and $KQ$ intersect at a point $J$ . Prove that $\triangle KJM$ is an isosceles triangle.