SHARYGIN 2015 Problem 13

Let $AH_1, BH_2$ and $CH_3$ be the altitudes of a triangle $ABC$ . Point $M$ is the midpoint of $H_2H_3$ . Line $AM$ meets $H_2H_1$ at point $K$ . Prove that $K$ lies on the medial line of $ABC$ parallel to $AC$ .