HMMT 2012 S6

Let $A B C$ be a triangle with $A B<A C$. Let the angle bisector of $\angle A$ and the perpendicular bisector of $B C$ intersect at $D$. Then let $E$ and $F$ be points on $A B$ and $A C$ such that $D E$ and $D F$ are perpendicular to $A B$ and $A C$, respectively. Prove that $B E=C F$.