# Distances to three incenters

266 views
0
Let $ABC$ be a triangle in which the angle bisector of $\angle{BAC}$ meets side $BC$ at $D$. Suppose that the distances from $A$ to the incenters of $\triangle{ABD}$, $\triangle{ACD}$, and $\triangle{ABC}$ are $16$, $18$, and $20$, not necessarily in order. Compute $AD$.