SHARYGIN 2016 Problem 3

Let $AH_1, BH_2$ be two altitudes of an acute-angled triangle $ABC$, $D$ be the projection of $H_1$ to $AC$, $E$ be the projection of $D$ to $AB$, $F$ be the common point of $ED$ and $AH_1$. Prove that $H_2F \parallel BC$.