Question 48
Let $ABC$ be a triangle and let $P$ be a point in the interior of the side $BC$. Let $I_{1}$ and $I_{2}$ be the incenters of the triangles $APB$ and $APC$, respectively. Let $X$ be the closest point to $A$ on the line $AP$ such that $XI_{1}$ is perpendicular to $XI_{2}$. Prove that the distance $AX$ is independent of the choice of $P$.