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)