KöMaL 2017 B4893

In a triangle $ABC$, $AB\ne BC$. The angle bisector drawn from point $B$ intersects side $AC$ at point $D$, and intersects the circumscribed circle again at point $E$. The circle of diameter $DE$ intersects the circumscribed circle again at a point $F$, different from $E$. Prove that the reflection of line $BF$ about the line $BD$ results in a median of triangle $ABC$.