BALKAN-SL 2020 G5

Let $ABC$ be an isosceles triangle with $AB = AC$ and $\angle A = 45^o$ . Its circumcircle $(c)$ has center $O, M$ is the midpoint of $BC$ and $D$ is the foot of the perpendicular from $C$ to $AB$ . With center $C$ and radius $CD$ we draw a circle which internally intersects $AC$ at the point $F$ and the circle $(c)$ at the points $Z$ and $E$ , such that $Z$ lies on the small arc $BC$ and $E$ on the small arc $AC$ . Prove that the lines $ZE$ , $CO$ , $FM$ are concurrent.

Brazitikos Silouanos, Greece