KöMaL 2019 B5057

Let $D$ and $E$ be points on leg $BC$ of a right-angled triangle with hypotenuse $AB$ such that $\angle DAC = \angle EAD= \angle BAE$. The feet of the perpendiculars dropped from vertex $C$ onto the line segment $AD$ and from point $D$ onto the hypotenuse $AB$ are $F$ and $K$, respectively. Line $CK$ intersects line segment $AE$ at point $H$, and the parallel line drawn from point $H$ to the line $AD$ intersects line segment $BC$ at point $M$. Show that point $F$ is the circumcentre of triangle $CHM$.