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 $1\le a\le b\le c<a+b$ and $a+b+c\le 200$.
Give your answer rounded to two digits after the decimal point.