Point $M$ is chosen in triangle $ABC$ so that the radii of the circumcircles of triangles $AMC, BMC$, and $BMA$ are no smaller than the radius of the circumcircle of $ABC$. Prove that all four radii are equal.
Problem
Source: Tournament of Towns Spring 2003 - Senior O-Level - Problem 3
Tags: geometry, circumcircle, geometry proposed