Question 96
Let $ABC$ be an acute triangle. Let $D$, $E$, and $F$ be the feet of altitudes from $A$, $B$, and $C$ to sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$, respectively, and let $Q$ be the foot of altitude from $A$ to line $EF$. Given that $AQ = 20$, $BC = 15$, and $AD = 24$, compute the perimeter of triangle $DEF$.