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Polish MO Final 2010, 3rd problem (parallelogram and circle)

Source:

November 7, 2010
geometryparallelogramcircumcircletrigonometrytrapezoidtrig identitiesLaw of Sines

Problem Statement

ABCDABCD is a parallelogram in which angle DABDAB is acute. Points A,P,B,DA, P, B, D lie on one circle in exactly this order. Lines APAP and CDCD intersect in QQ. Point OO is the circumcenter of the triangle CPQCPQ. Prove that if D≠OD \neq O then the lines ADAD and DODO are perpendicular.
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