NL-NWO 2009 Problem 4

Let $ABC$ be an arbitrary triangle. On the perpendicular bisector of $AB$ , there is a point $P$ inside of triangle $ABC$ . On the sides $BC$ and $CA$ , triangles $BQC$ and $CRA$ are placed externally. These triangles satisfy $\vartriangle BPA \sim \vartriangle BQC \sim \vartriangle CRA$ . (So $Q$ and $A$ lie on opposite sides of $BC$ , and $R$ and $B$ lie on opposite sides of $AC$ .) Show that the points $P, Q, C$ and $R$ form a parallelogram.