ELMO SL 2014 G11

Let $ABC$ be a triangle with circumcenter $O$ . Let $P$ be a point inside $ABC$ , so let the points $D, E, F$ be on $BC, AC, AB$ respectively so that the Miquel point of $DEF$ with respect to $ABC$ is $P$ . Let the reflections of $D, E, F$ over the midpoints of the sides that they lie on be $R, S, T$ . Let the Miquel point of $RST$ with respect to the triangle $ABC$ be $Q$ . Show that $OP = OQ$ .

Proposed by Yang Liu