SHARYGIN 2020 Problem 12

Let $H$ be the orthocenter of a nonisosceles triangle $ABC$ . The bisector of angle $BHC$ meets $AB$ and $AC$ at points $P$ and $Q$ respectively. The perpendiculars to $AB$ and $AC$ from $P$ and $Q$ meet at $K$ . Prove that $KH$ bisects the segment $BC$ .