SHARYGIN 2018 Problem 15

The altitudes $AH_1,BH_2,CH_3$ of an acute-angled triangle $ABC$ meet at point $H$ . Points $P$ and $Q$ are the reflections of $H_2$ and $H_3$ with respect to $H$ . The circumcircle of triangle $PH_1Q$ meets for the second time $BH_2$ and $CH_3$ at points $R$ and $S$ . Prove that $RS$ is a medial line of triangle $ABC$ .