Question 38
Let $\triangle ABC$ be a triangle inscribed in a circle $\Gamma$ with center $O$. Let $I$ be the incenter of $\triangle ABC$ and $D, E, F$ the contact points of the incircle of $\triangle ABC$ with $BC, AC, AB$, respectively. If $S$ is the foot of the perpendicular from $D$ to the line $EF$, prove that the line $SI$ passes through the antipodal of $A$ with respect to $\Gamma$.