## Triangle of Circular Arcs

### Problem 727

Let $r_a$, $r_b$ and $r_c$ be the radii of three circles that are mutually and externally tangent to each other. The three circles then form a triangle of circular arcs between their tangency points as shown for the three blue circles in the picture below.

Let $\mathbb{E}(d)$ be the expected value of $d$ when $r_a$, $r_b$ and $r_c$ are integers chosen uniformly such that $1\leq r_a<r_b<r_c \leq 100$ and $\text{gcd}(r_a,r_b,r_c)=1$.

Find $\mathbb{E}(d)$, rounded to eight places after the decimal point.