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Prove perpendicularity

Source: Mexico National Olympiad Mock Exam 2021 P3

November 12, 2021
geometry

Problem Statement

Let PP and QQ be points in the interior of a triangle ABCABC such that ∠APC=∠AQB=90∘\angle APC = \angle AQB = 90^{\circ}, ∠ACP=∠PBC\angle ACP = \angle PBC, and ∠ABQ=∠QCB\angle ABQ = \angle QCB. Suppose that lines BPBP and CQCQ meet at a point RR. Show that ARAR is perpendicular to PQPQ.
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