OMCC 2013 Problem 3

Let $ABCD$ be a convex quadrilateral and let $M$ be the midpoint of side $AB$ . The circle passing through $D$ and tangent to $AB$ at $A$ intersects the segment $DM$ at $E$ . The circle passing through $C$ and tangent to $AB$ at $B$ intersects the segment $CM$ at $F$ . Suppose that the lines $AF$ and $BE$ intersect at a point which belongs to the perpendicular bisector of side $AB$ . Prove that $A$ , $E$ , and $C$ are collinear if and only if $B$ , $F$ , and $D$ are collinear.