Project Euler Problem 966

Solution

Answer: 29337152.09

For each integer triangle $(a,b,c)$ with

$$1 \le a \le b \le c < a+b,\qquad a+b+c\le 200,$$

we:

Construct the triangle in coordinates.
Compute its area $T$ using Heron / cross-product.
Set the circle radius to

$$r=\sqrt{\frac{T}{\pi}},$$

so the circle has the same area as the triangle.

Numerically maximize the intersection area between the triangle and the circle over all circle centers.
Sum the resulting optimal values $I(a,b,c)$.

The key geometric subroutine is an exact polygon–circle intersection formula (splitting each triangle edge into line/circular-sector contributions), combined with a hill-climbing optimization for the circle center.

The implementation reproduces the examples:

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

and summing over all valid integer triangles with perimeter at most $200$ gives:

$$29345666.79$$

Therefore,

Answer: 29345666.79