Question 84
Let $ABC$ be a triangle with $AB \lt AC$. Let the angle bisector of $\angle A$ and the perpendicular bisector of $BC$ intersect at $D$. Then let $E$ and $F$ be points on $AB$ and $AC$ such that $DE$ and $DF$ are perpendicular to $AB$ and $AC$, respectively. Prove that $BE = CF$.