Question 85
The incircle of a triangle $ABC$ touches its sides $BC$, $CA$, $AB$ at the points $A_1$, $B_1$, $C_1$, respectively. Let the projections of the orthocenter $H_1$ of the triangle $A_1B_1C_1$ to the lines $AA_1$ and $BC$ be $P$ and $Q$, respectively. Show that the line $PQ$ bisects the line segment $B_1C_1$.