Question 109
Let $ABC$ be a triangle, and $I$ its incenter. Let the incircle of $ABC$ touch side $BC$ at $D$, and let lines $BI$ and $CI$ meet the circle with diameter $AI$ at points $P$ and $Q$, respectively. Given $BI = 6$, $CI = 5$, $DI = 3$, determine the value of $(DP / DQ)^2$.