Problem 966

Triangle Circle Intersection

Let $I(a, b, c)$ be the largest possible area of intersection between a triangle of side lengths $a, b, c$ and a circle which has the same area as the triangle.

For example $I(3, 4, 5) \approx 4.593049$ and $I(3, 4, 6) \approx 3.552564$.

Find the sum of $I(a, b, c)$ for integers $a, b, c$ such that $1 \le a \le b \le c \lt a + b$ and $a + b + c \le 200$.

Give your answer rounded to two digits after the decimal point.

三角形与圆的交集

记$I(a, b, c)$为边长为$a, b, c$的三角形与一个面积与之相等的圆之间的最大交集面积。

例如，$I(3, 4, 5) \approx 4.593049$，$I(3, 4, 6) \approx 3.552564$。

对于满足$1 \le a \le b \le c \lt a + b$且$a + b + c \le 200$的整数$a, b, c$，求所有$I(a, b, c)$的和，并四舍五入到小数点后两位作为你的答案。