A triangle with sides $8,15,17$ has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles?

1. Place the right angle at $C=(0,0)$ with legs $CA=15$ and $CB=8$ on the axes. The circumcenter is the midpoint of the hypotenuse, $O=\left(\frac{15}2,4\right)$, and the inradius is $r=\frac{8+15-17}2=3$, so $I=(3,3)$.
2. (Equivalently, Euler's formula $OI^2=R(R-2r)$ with $R=\frac{17}2$ gives the same value.) Directly, $OI^2=\left(\frac92\right)^2+1^2=\frac{85}4$.
3. $OI=\boxed{\dfrac{\sqrt{85}}2}$.