For C large: "The smallest difference in angle between two points is approximately 1.25 * 10-13 radians." It looks like ( - n,  - n) ( - n + 1,  - n + 1) and (n, n - 1) form an optimal angle, but why ?

Here is a quick proof: minimizing the angle is equivalent to minimizing its sine. And then the area of a triangle with sides of length b and c is 2 * b * c * sin(A)

By Pick's Theorem: the smallest area of a triangle with integer coordinates is 1/2. It's the case when no points lie inside or on the edges. So, what remains is to take the two largest segments with no lattice points inside them.