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Midpoint on diagonal of cyclic quadrilateral

Source: Mexico National Olympiad 2017, Problem 5

November 7, 2017
geometrycyclic quadrilateral

Problem Statement

On a circle Γ\Gamma, points A,B,N,C,D,MA, B, N, C, D, M are chosen in a clockwise order in such a way that NN and MM are the midpoints of clockwise arcs BCBC and ADAD respectively. Let PP be the intersection of ACAC and BDBD, and let QQ be a point on line MBMB such that PQPQ is perpendicular to MNMN. Point RR is chosen on segment MCMC such that QB=RCQB = RC, prove that the midpoint of QRQR lies on ACAC.
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