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right angle wanted, collinear given (Kharkiv City XI 2014 - Ukr)

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March 14, 2020
right anglegeometrycollinearperpendicularKharkiv

Problem Statement

In the convex quadrilateral of the ABCDABCD, the diagonals of ACAC and BDBD are mutually perpendicular and intersect at point EE. On the side of ADAD, a point PP is chosen such that PE=ECPE = EC. The circumscribed circle of the triangle BCDBCD intersects the segment ADAD at the point QQ. The circle passing through point AA and tangent to the line EPEP at point PP intersects the segment ACAC at point RR. It turns out that points B,Q,RB, Q, R are collinear. Prove that ∠BCD=90o\angle BCD = 90^o.
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