The points \(A\), \(B\) and \(C\) lie on the surface of a sphere with center \(O\) and radius \(20\). It is given that \(AB=13\), \(BC=14\), \(CA=15\), and that the distance from \(O\) to triangle \(ABC\) is \(\frac{m\sqrt{n}}k\), where \(m\), \(n\), and \(k\) are positive integers, \(m\) and \(k\) are relatively prime, and \(n\) is not divisible by the square of any prime. Find \(m+n+k\).
(第十八届AIME2 2000 第12题)