The nine-point-circle

Geometry Level 5

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

  • the midpoint of each side,
  • the foot of each altitude, and
  • the midpoint of the line segment from each vertex to the orthocenter.

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

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


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