$ABC$ is an acute triangle and $AD$, $BE$ and $CF$ are the altitudes, with $H$ being the point of intersection of these altitudes. Points $A_1$, $B_1$, $C_1$ are chosen on rays $AD$, $BE$ and $CF$ respectively such that $AA_1 = HD$, $BB_1 = HE$ and $CC_1 =HF$. Let $A_2$, $B_2$ and $C_2$ be midpoints of segments $A_1D$, $B_1E$ and $C_1F$ respectively. Prove that $H$, $A_2$, $B_2$ and $C_2$ are concyclic. (Mykhailo Barkulov)