The nine-point-circle

Geometry Level 4

For a triangle with sides 13, 15 and altitude 12, find the radius of the circle that passes through the following points:

If the radius can be written as $a + \frac{m}{n}$ , where $a,m,$ and $n$ are positive integers, $m<n,$ and $m$ and $n$ are coprime, find $a+m+n$ .

Note: Assume the given altitude to be through the vertex common to both the given sides.