China Girls' Mathematical Olympiad

Suppose that the convex quadrilateral $ABCD$ satisfies $AB=BC$, $AD=DC$. $E$ is a point on $AB$, and $F$ on $AD$, such that $B$, $E$, $F$, $D$ are concyclic. Draw $△DPE$ directly similar to $△ADC$, and $△BQF$ directly similar to $△ABC$. Prove that $A$, $P$, $Q$ are collinear. (Posed by Ye Zhonghao)

Denote by $O$ the center of the circle that passes through $B$, $E$, $F$, $D$. Draw lines $OB$, $OF$, $BD$.