KMO 2009 Problem 1

Let $I, O$ be the incenter and the circumcenter of triangle $ABC$ , and $D,E,F$ be the circumcenters of triangle $BIC, CIA, AIB$ . Let $P, Q, R$ be the midpoints of segments $DI, EI, FI$ . Prove that the circumcenter of triangle $PQR$ , $M$ , is the midpoint of segment $IO$ .