Question 8
Let $ABC$ be a triangle with incenter $I$ and let $\Gamma$ be a circle with center $I$, with radius greater than that of the incircle and not passing through any of the vertices. Let $X_1$ be the intersection point of $\Gamma$ with line $AB$ closer to $B$; $X_2$, $X_3$ the intersection points of $\Gamma$ with line $BC$, with $X_2$ closer to $B$; and $X_4$ the intersection point of $\Gamma$ with line $CA$ closer to $C$. Let $K$ be the intersection point of lines $X_1X_2$ and $X_3X_4$. Prove that $AK$ meets segment $X_2X_3$ at its midpoint.