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Easy geometry with lines intersecting circles

Source: Latvian TST for Baltic Way 2022 P9

November 24, 2022
geometrycyclic quadrilateralcircumcircle

Problem Statement

Let ABCDABCD be a cyclic quadrilateral inscribed in circle Ω\Omega. Let the lines ABAB and CDCD intersect at PP, and the lines ADAD and BCBC intersect at QQ. Let then the circumcircle of the triangle △APQ\triangle APQ intersect Ω\Omega at R≠AR \neq A. Prove that the line CRCR goes through the midpoint of the segment PQPQ.
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