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AM = AB if (BMC) and (AXY) are tangent

Source: 2022 South Russian Girls MO - Assara Seniors p8

March 1, 2024
geometrytangent circles

Problem Statement

In parallelogram ABCDABCD, angle AA is acute. Let XX be a point, symmetrical to point CC wrt to straight line ADAD, YY is a point symmetrical to the point CC wrt point DD, and MM is the intersection point of ACAC and BDBD. It turned out, that the circumcircles of triangles BMCBMC and AXYAXY are tangent internally. Prove that AM=ABAM = AB.
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