MathNet 2012 Question 77

Let $ABC$ be a non-isosceles triangle with circumcenter $U$ and incenter $I$. Assume that the bisector of the segment $UI$ passes through the common point of the angle bisector of $\gamma = \angle ACB$ with the circumcircle of $ABC$. Prove that $\gamma$ is the second largest angle in the triangle $ABC$.
G. Baron, Vienna