Question 94
Let $ABC$ be a triangle with $AB = 5$, $BC = 8$, $CA = 11$. The incircle $\omega$ and $A$-excircle $\Gamma$ are centered at $I_1$ and $I_2$, respectively, and are tangent to $BC$ at $D_1$ and $D_2$, respectively. Find the ratio of the area of $\triangle A I_1 D_1$ to the area of $\triangle A I_2 D_2$.