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Prove that all four radii are equal

Source: Tournament of Towns Spring 2003 - Senior O-Level - Problem 3

June 14, 2011
geometrycircumcirclegeometry proposed

Problem Statement

Point MM is chosen in triangle ABCABC so that the radii of the circumcircles of triangles AMC,BMCAMC, BMC, and BMABMA are no smaller than the radius of the circumcircle of ABCABC. Prove that all four radii are equal.
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