Question 5
Let $ABCD$ be a fixed convex quadrilateral with $BC=DA$ and $\overline{BC} \nparallel \overline{DA}$. Let two variable points $E$ and $F$ lie on 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 $P$.