KöMaL 2018 B4987

The circumcentre of an acute-angled scalene triangle $ABC$ is $O$, its orthocentre is $M$, the foot of the altitude drawn from vertex $A$ is $D$, and the midpoint of side $AB$ is $F$. The ray drawn from $F$ through $M$ intersects the circumcircle of triangle $ABC$ at $G$. $a)$ Prove that the points $A$, $F$, $D$ and $G$ are concyclic. $b)$ Let $K$ denote the circle in $a)$, and let $E$ be the midpoint of line segment $CM$. Prove that $EK=OK$.