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two lines concurrent with a circumcircle

Source: 2019 Oral Moscow Geometry Olympiad grades 10-11 p5

May 22, 2019
geometrycircumcirclecyclic quadrilateralconcurrent

Problem Statement

On sides ABAB and BCBC of a non-isosceles triangle ABCABC are selected points C1C_1 and A1A_1 such that the quadrilateral AC1A1CAC_1A_1C is cyclic. Lines CC1CC_1 and AA1AA_1 intersect at point PP. Line BPBP intersects the circumscribed circle of triangle ABCABC at the point QQ. Prove that the lines QC1QC_1 and CMCM, where MM is the midpoint of A1C1A_1C_1, intersect at the circumscribed circles of triangle ABCABC.
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