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concyclic with two vertices and the circumcenter

Source: MEMO 2016 I3

August 24, 2016
geometrygeometry proposedCircumcenterperpendicular lines

Problem Statement

Let ABCABC be an acute triangle such that ∠BAC>45∘\angle BAC > 45^{\circ} with circumcenter OO. A point PP is chosen inside triangle ABCABC such that A,P,O,BA, P, O, B are concyclic and the line BPBP is perpendicular to the line CPCP. A point QQ lies on the segment BPBP such that the line AQAQ is parallel to the line POPO.
Prove that ∠QCB=∠PCO\angle QCB = \angle PCO.
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