DE-BUNDESWETTBEWERB 2021 Problem 7

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 $\angle BFC$ .