IR-TST 2019 Problem 4

Consider triangle $ABC$ with orthocenter $H$ . Let points $M$ and $N$ be the midpoints of segments $BC$ and $AH$ . Point $D$ lies on line $MH$ so that $AD\parallel BC$ and point $K$ lies on line $AH$ so that $DNMK$ is cyclic. Points $E$ and $F$ lie on lines $AC$ and $AB$ such that $\angle EHM=\angle C$ and $\angle FHM=\angle B$ . Prove that points $D,E,F$ and $K$ lie on a circle.

Proposed by Alireza Dadgarnia