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Perpendiculars, Parallels and six points on a circle

Source: Bundeswettbewerb Mathematik 2018, Round 1 - #3

June 30, 2018
geometryorthocenterperpendicularparallelcircle

Problem Statement

Let HH be the orthocenter of the acute triangle ABCABC. Let HaH_a be the foot of the perpendicular from AA to BCBC and let the line through HH parallel to BCBC intersect the circle with diameter AHaAH_a in the points PaP_a and QaQ_a. Similarly, we define the points Pb,QbP_b, Q_b and Pc,QcP_c,Q_c.
Show that the six points Pa,Qa,Pb,Qb,Pc,QcP_a,Q_a,P_b,Q_b,P_c,Q_c lie on a common circle.
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