jmc

The triangle is a right triangle that can be placed in a coordinate system with vertices at $(0,0)$, $(5,0)$, and ($0,12)$. The center of the circumscribed circle is the midpoint of the hypotenuse, which is $(5/2,6)$.



To determine the radius $r$ of the inscribed circle notice that the hypotenuse of the triangle is $$(12-r)+(5-r)=13$$so $r=2$.

So the center of the inscribed circle is $(2,2)$, and the distance between the two centers is $$\sqrt{{\left(\frac{5}{2}-2\right)}^2+(6-2{)}^2}=\boxed{\frac{\sqrt{65}}{2}}.$$