OMM 2005 Problem 1

Let $O$ be the center of the circumcircle of an acute triangle $ABC$ , let $P$ be any point inside the segment $BC$ . Suppose the circumcircle of triangle $BPO$ intersects the segment $AB$ at point $R$ and the circumcircle of triangle $COP$ intersects $CA$ at point $Q$ . (i) Consider the triangle $PQR$ , show that it is similar to triangle $ABC$ and that $O$ is its orthocenter. (ii) Show that the circumcircles of triangles $BPO$ , $COP$ , $PQR$ have the same radius.