Consider an acute triangle $ABC$ in which $A_1, B_1,$ and $C_1$ are the feet of the altitudes from $A, B,$ and $C,$ respectively, and $H$ is the orthocenter. The perpendiculars from $H$ onto $A_1C_1$ and $A_1B_1$ intersect lines $AB$ and $AC$ at $P$ and $Q,$ respectively. Prove that the line perpendicular to $B_1C_1$ that passes through $A$ also contains the midpoint of the line segment $PQ$.