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diagonal intersection of cyclic ABCD is incenter of BMC, circumcircles related

Source: 2013 Oral Moscow Geometry Olympiad grades 10-11 p1

August 16, 2019
geometryincentercircumcirclecyclic quadrilateral

Problem Statement

Diagonals of a cyclic quadrilateral ABCDABCD intersect at point OO. The circumscribed circles of triangles AOBAOB and CODCOD intersect at point MM on the side ADAD. Prove that the point OO is the center of the inscribed circle of the triangle BMCBMC.