Question 31
A circle $k$ through the vertices $A$ and $B$ of an acute $\triangle ABC$ meets the sides $AC$ and $BC$ at inner points $M$ and $N$, respectively. The tangent lines to $k$ at the points $M$ and $N$ meet at point $O$. Prove that $O$ is the circumcenter of $\triangle CMN$ if and only if $AB$ is a diameter of $k$.