KMO 2017 Problem 6

In a quadrilateral $ABCD$ , we have $\angle ACB = \angle ADB = 90$ and $CD < BC$ . Denote $E$ as the intersection of $AC$ and $BD$ , and let the perpendicular bisector of $BD$ hit $BC$ at $F$ . The circle with center $F$ which passes through $B$ hits $AB$ at $P (\neq B)$ and $AC$ at $Q$ . Let $M$ be the midpoint of $EP$ . Prove that the circumcircle of $EPQ$ is tangent to $AB$ if and only if $B, M, Q$ are colinear.