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AR = QR wanted inside a regular pentagon (2019 Kyiv City MO Round2 10.3.1)

Source:

September 18, 2020
geometryequal segmentspentagonregular pentagon

Problem Statement

Let ABCDEABCDE be a regular pentagon with center MM. Point P≠MP \ne M is selected on segment MDMD. The circumscribed circle of triangle ABPABP intersects the line AEAE for second time at point QQ, and a line that is perpendicular to the CDCD and passes through PP, for second time at the point RR. Prove that AR=QRAR = QR.
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