Perpendicular bisectors of the sides $BC$ and $AC$ of an acute-angled triangle $ABC$ intersect lines $AC$ and $BC$ at points $M$ and $N$. Let point $C$ move along the circumscribed circle of triangle $ABC$, remaining in the same half-plane relative to $AB$ (while points $A$ and $B$ are fixed). Prove that line $MN$ touches a fixed circle.
Problem
Source: 2010 Oral Moscow Geometry Olympiad grades 8-9 p6
Tags: geometry, perpendicular bisector, circumcircle, fixed, circle