Question 13
Let $D$ be a variable point on side $BC$ of triangle $ABC$. The incenters of triangles $ABC$, $ABD$ and $ACD$ are $I$, $I_1$ and $I_2$, respectively. Points $M$ and $N$ are the second intersection points of circumcircles of triangles $IAI_1$ and $IAI_2$ with the circumcircle of triangle $ABC$, respectively. Prove that as $D$ varies on side $BC$, line $MN$ passes through a fixed point in the plane.