PL-OM 2004 Problem 1

A point $D$ is taken on the side $AB$ of a triangle $ABC$ . Two circles passing through $D$ and touching $AC$ and $BC$ at $A$ and $B$ respectively intersect again at point $E$ . Let $F$ be the point symmetric to $C$ with respect to the perpendicular bisector of $AB$ . Prove that the points $D,E,F$ lie on a line.