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 $MD$.
Školjka 
be an acute-angled triangle with 
, circumcentre 
and circumcircle 
. Let the tangents to 
and 
meet each other at 
, and let the line 
intersect 
at 
. Denote the midpoint of 
and let 
meet 
. Finally, let 
be a point on 
and 
are concyclic. 
bisects the segment 
.