We are given a circle $k$ and a point $A$ outside of $k$. Next we draw three lines through $A$: one secant intersecting the circle $k$ at points $B$ and $C$, and two tangents touching the circle$k$ at points $D$ and $E$. Let $F$ be the midpoint of $DE$. Show that the line $DE$ bisects the angle $\angle BFC$.
Problem
Source: Bundeswettbewerb Mathematik 2021, Round 2 - Problem 3
Tags: geometry, geometry proposed, tangent, angle bisector