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BM = KQ wanted, MK || AB, 2 circles related (2012 Kyiv City MO 11.3)

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August 1, 2020
geometryequal segmentsparallelcircles

Problem Statement

Inside the triangle ABCABC choose the point MM, and on the side BCBC - the point KK in such a way that MK∣∣ABMK || AB. The circle passing through the points M,  K,  C,M, \, \, K, \, \, C, crosses the side ACAC for the second time at the point NN, a circle passing through the points M,  N,  A,M, \, \, N, \, \, A, crosses the side ABAB for the second time at the point QQ. Prove that BM=KQBM = KQ.
(Nagel Igor)
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