Problem 727

Triangle of Circular Arcs

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.

Define the circumcircle of this triangle to be the red circle, with centre $D$ , passing through their tangency points. Further define the incircle of this triangle to be the green circle, with centre $E$ , that is mutually and externally tangent to all the three blue circles. Let $d = | D E |$ be the distance between the centres of the circumcircle and the incircle.

Let $E (d)$ 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 $gcd (r_{a}, r_{b}, r_{c}) = 1$ .

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

圆弧三角形

三个半径为 $r_{a}$ 、 $r_{b}$ 和 $r_{c}$ 的圆两两外切，在切点之间构成了一个圆弧三角形，如下图的三个蓝色圆所示。

记圆弧三角形的外接圆为上图中的红色圆，圆心点为 $D$ 且过三个切点。再记圆弧三角形的内切圆为上图中的绿色圆，圆心点为 $E$ 且与三个蓝色圆外切。记 $d = | D E |$ 为外接圆和内切圆的圆心距。

若 $r_{a}$ 、 $r_{b}$ 和 $r_{c}$ 均为在 $1 \leq r_{a} < r_{b} < r_{c} \leq 100$ 范围内随机均匀抽取的整数且满足 $gcd (r_{a}, r_{b}, r_{c}) = 1$ ，记 $E (d)$ 为 $d$ 的期望值。

求 $E (d)$ 并保留小数点后八位数字。