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