2008 AIME II Problem 11

In triangle A B C, A B=A C=100, and B C=56. Circle P has radius 16 and is tangent to \overline{A C} and
\overline{B C}. Circle Q is externally tangent to circle P and is tangent to \overline{A B} and
\overline{B C}. No point of circle Q lies outside of \triangle A B C. 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+n k.