CGMO 2008 Problem 5

In convex quadrilateral $ABCD$ , $AB = BC$ and $AD = DC$ . Point $E$ lies on segment $AB$ and point $F$ lies on segment $AD$ such that $B$ , $E$ , $F$ , $D$ lie on a circle. Point $P$ is such that triangles $DPE$ and $ADC$ are similar and the corresponding vertices are in the same orientation (clockwise or counterclockwise). Point $Q$ is such that triangles $BQF$ and $ABC$ are similar and the corresponding vertices are in the same orientation. Prove that points $A$ , $P$ , $Q$ are collinear.