In triangle $ABC$, $AB = AC = 100$, and $BC = 56$. [Circle](https://artofproblemsolving.com/wiki/index.php/Circle) $P$ has [radius](https://artofproblemsolving.com/wiki/index.php/Radius) $16$ and is tangent to $\overline{AC}$ and $\overline{BC}$. Circle $Q$ is externally [tangent](https://artofproblemsolving.com/wiki/index.php/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 $m + nk$.
Let $X$ and $Y$ be the feet of the [perpendiculars](https://artofproblemsolving.com/wiki/index.php/Perpendicular) from $P$ and $Q$ to $BC$, respectively. Let the radius of $\odot Q$ be $r$. We know that $PQ = r + 16$. From $Q$ draw segment $\overline{QM} \parallel \overline{BC}$ such that $M$ is on $PX$. Clearly, $QM = XY$ and $PM = 16-r$. Also, we know $QPM$ is a right triangle.
To find $XC$, consider the right triangle $PCX$. Since $\odot P$ is tangent to $\overline{AC},\overline{BC}$, then $PC$ [bisects](https://artofproblemsolving.com/wiki/index.php/Bisect) $\angle ACB$. Let $\angle ACB = 2\theta$; then $\angle PCX = \angle QBX = \theta$. Dropping the altitude from $A$ to $BC$, we recognize the $7 - 24 - 25$ [right triangle](https://artofproblemsolving.com/wiki/index.php/Right_triangle), except scaled by $4$.
So we get that $\tan(2\theta) = 24/7$. From the [half-angle identity](https://artofproblemsolving.com/wiki/index.php/Trigonometric_identity), we find that $\tan(\theta) = \frac {3}{4}$. Therefore, $XC = \frac {64}{3}$. By similar reasoning in triangle $QBY$, we see that $BY = \frac {4r}{3}$.
We conclude that $XY = 56 - \frac {4r + 64}{3} = \frac {104 - 4r}{3}$.
So our right triangle $QPM$ has sides $r + 16$, $r - 16$, and $\frac {104 - 4r}{3}$.
By the [Pythagorean Theorem](https://artofproblemsolving.com/wiki/index.php/Pythagorean_Theorem), simplification, and the [quadratic formula](https://artofproblemsolving.com/wiki/index.php/Quadratic_formula), we can get $r = 44 - 6\sqrt {35}$, for a final answer of $\fbox{254}$.