OIM 2007 Problem 2

Let $ABC$ be a triangle with incenter $I$ and let $\Gamma$ be a circle centered at $I$ , whose radius is greater than the inradius and does not pass through any vertex. Let $X_{1}$ be the intersection point of $\Gamma$ and line $AB$ , closer to $B$ ; $X_{2}$ , $X_{3}$ the points of intersection of $\Gamma$ and line $BC$ , with $X_{2}$ closer to $B$ ; and let $X_{4}$ be the point of intersection of $\Gamma$ with line $CA$ closer to $C$ . Let $K$ be the intersection point of lines $X_{1}X_{2}$ and $X_{3}X_{4}$ . Prove that $AK$ bisects segment $X_{2}X_{3}$ .