Eccoci.

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.