## Problem 1.

Let $M,\,N$ two points interior to the circle $\mathcal{C}(O)$ such that $O$ is the midpoint of $MN.$ Let $S$ an arbitrary point lies on $\mathcal C(O),$ and $E,\,F$ the second intersections of the lines $SM,\,SN$ with $\mathcal C(O),$ respectively. The tangents in $E,\,F$ with respect to the circle $\mathcal C(O)$ intersect each other at $I.$ Prove that the perpendicular bisector of the segment $MN$ passes through the midpoint of $SI.$

(Mathematical Reflections 2007)