Challenge Your Brain 2

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 ${P}$ 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 ${m} + {n}{k}$ .