A sphere with center $O$ has radius $5$. A triangle with sides $13$, $14$, and $15$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?

1. The sphere meets the plane of the triangle in a circle, and this circle is tangent to all three sides, so it is the incircle. Its center is the foot of the perpendicular from $O$ to the plane.
2. The triangle $13,14,15$ has $s=21$ and area $\sqrt{21\cdot8\cdot7\cdot6}=84$, so the inradius is $\rho=\frac{84}{21}=4$.
3. The distance from $O$ to the plane satisfies $d^2+\rho^2=5^2$, so $d=\sqrt{25-16}=\boxed{3}$.