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Configuration with three similar triangles

Source: IGO 2022 Intermediate P3

December 13, 2022
geometrysimilar triangles

Problem Statement

Let OO be the circumcenter of triangle ABCABC. Arbitrary points MM and NN lie on the sides ACAC and BCBC, respectively. Points PP and QQ lie in the same half-plane as point CC with respect to the line MNMN, and satisfy △CMN∼△PAN∼△QMB\triangle CMN \sim \triangle PAN \sim \triangle QMB (in this exact order). Prove that OP=OQOP=OQ.
Proposed by Medeubek Kungozhin, Kazakhstan
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