2007 OIM Problems/Problem 2

Problem

Let $ABC$ be a triangle with incenter $I$ and $\Gamma$ be a circle with center $I$ , of radius greater than that of the inscribed circle and that does not pass through any of the vertices. Let $X_1$ be the point of intersection of $\Gamma$ with the line $AB$ closest to $B$ ; $X_2$ , $X_3$ the points of intersection of $\Gamma$ with the line $BC$ being $X_2$ closest to $B$ ; and $X_4$ the point of intersection of $\Gamma$ with the line $CA$ closest to $C$ . Let $K$ be the point of intersection of the lines $X_1X_2$ and $X_3X_4$ . Show that $AK$ intersects segment $X_2X_3$ at its midpoint.