MEMO 2017 Problem 6

Let $ABC$ be an acute-angled triangle with $AB \neq AC$ , circumcentre $O$ and circumcircle $\Gamma$ . Let the tangents to $\Gamma$ at $B$ and $C$ meet each other at $D$ , and let the line $AO$ intersect $BC$ at $E$ . Denote the midpoint of $BC$ by $M$ and let $AM$ meet $\Gamma$ again at $N \neq A$ . Finally, let $F \neq A$ be a point on $\Gamma$ such that $A, M, E$ and $F$ are concyclic. Prove that $FN$ bisects the segment $MD$ .