OME 2001 Problem 2

Let $P$ be a point on the interior of triangle $ABC$ , such that the triangle $ABP$ satisfies $AP = BP$ . On each of the other sides of $ABC$ , build triangles $BQC$ and $CRA$ exteriorly, both similar to triangle $ABP$ satisfying: $$BQ = QC$$ and $$CR = RA.$$ Prove that the point $P,Q,C,$ and $R$ are collinear or are the vertices of a parallelogram.