Question 11
In triangle $ABC,$ $AB = AC = 100,$ and $BC = 56.$ Circle $P$ has radius $16$ and is tangent to $\overline{AC}$ and $\overline{BC}.$ Circle $Q$ is externally tangent to circle $P$ and is tangent to $\overline{AB}$ and $\overline{BC}.$ No point of circle $Q$ lies outside of $\triangle ABC.$ The radius of circle $Q$ can be expressed in the form $m - n\sqrt{k},$ where $m,$ $n,$ and $k$ are positive integers and $k$ is the product of distinct primes. Find $m + nk.$