15. As shown in Figure 2, two equal circles with a radius of 5 are externally tangent to each other, and both are internally tangent to a larger circle with a radius of 13, with the points of tangency being $A$ and $B$. Let $AB = \frac{m}{n}\left(m, n \in \mathbf{Z}_{+},(m, n)\right. = 1)$. Then the value of $m+n$ is ( ).
(A) 21
(B) 29
(C) 58
(D) 69
(E) 93
15. D.

As shown in Figure 5, let the centers of the three circles be $X$, $Y$, and $Z$.

Then points $A$ and $B$ are on the extensions of $XY$ and $XZ$, respectively, satisfying
$$
\begin{array}{l}
XY = XZ \\
= 13 - 5 = 8, \\
YZ = 5 + 5 = 10, \\
XA = XB = 13.
\end{array}
$$

Since $YZ \parallel AB \Rightarrow \triangle XYZ \sim \triangle XAB$
$$
\Rightarrow \frac{XA}{AB} = \frac{XY}{YZ} \Rightarrow AB = \frac{XA \cdot YZ}{XY} = \frac{65}{4} \text{.}
$$

Thus, $m = 65$, $n = 4$, and $m + n = 69$.