Question 112
The circle $\Gamma_1$, with radius $r$, is internally tangent to the circle $\Gamma_2$ at $S$. The chord $AB$ of $\Gamma_2$ is tangent to $\Gamma_1$ at $C$. Let $M$ be the midpoint of the arc $\widehat{AB}$ (not containing $S$), and let $N$ be the foot of the perpendicular from $M$ to the line $AB$. Prove that $AC \times CB = 2r \times MN$. (Posed by Ye Zhonghao)