DE-BUNDESWETTBEWERB 2018 Problem 3

Let $H$ be the orthocenter of the acute triangle $ABC$ . Let $H_a$ be the foot of the perpendicular from $A$ to $BC$ and let the line through $H$ parallel to $BC$ intersect the circle with diameter $AH_a$ in the points $P_a$ and $Q_a$ . Similarly, we define the points $P_b, Q_b$ and $P_c,Q_c$ .

Show that the six points $P_a,Q_a,P_b,Q_b,P_c,Q_c$ lie on a common circle.