Let be an acute-angled triangle with , circumcentre and circumcircle . Let the tangents to at and meet each other at , and let the line intersect at . Denote the midpoint of by and let meet again at . Finally, let be a point on such that and are concyclic. Prove that bisects the segment .
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 $MD$.
Source: Srednjoeuropska matematička olimpijada 2017, ekipno natjecanje, problem 6