Question 12
Let $I$ be the incenter of triangle $\triangle ABC$, and $D$ be an arbitrary point on arc $\widehat{AB}$, not containing point $C$, of the circumcircle of $\triangle ABC$. Let $I_A$ and $I_B$ be the incenters of $\triangle BCD$ and $\triangle ACD$ respectively. Lines $DI_A$ and $II_B$ intersect at point $X_A$, and lines $DI_B$ and $II_A$ intersect at point $X_B$. Prove that the intersection of lines $X_A X_B$ and $I_A I_B$ lies on a fixed line, independent of the choice of $D$.