IMO 2005 Problem 5

Let $ABCD$ be a fixed convex quadrilateral with $BC=DA$ and $BC$ not parallel with $DA$ . Let two variable points $E$ and $F$ lie of the sides $BC$ and $DA$ , respectively and satisfy $BE=DF$ . The lines $AC$ and $BD$ meet at $P$ , the lines $BD$ and $EF$ meet at $Q$ , the lines $EF$ and $AC$ meet at $R$ .

Prove that the circumcircles of the triangles $PQR$ , as $E$ and $F$ vary, have a common point other than $P$ .