Question 138
Triangle $ABC$ has incircle $\omega$ which touches $AB$ at $C_1$, $BC$ at $A_1$, and $CA$ at $B_1$. Let $A_2$ be the reflection of $A_1$ over the midpoint of $BC$, and define $B_2$ and $C_2$ similarly. Let $A_3$ be the intersection of $AA_2$ with $\omega$ that is closer to $A$, and define $B_3$ and $C_3$ similarly. If $AB=9$, $BC=10$, and $CA=13$, find $\left[A_3 B_3 C_3\right] / [ABC]$. (Here $[XYZ]$ denotes the area of triangle $XYZ$.)