Triangle on parabola

On the parabola y = x²/k, three points A(a, a²/k), B(b, b²/k) and C(c, c²/k) are chosen.

Let F(K, X) be the number of the integer quadruplets (k, a, b, c) such that at least one angle of the triangle ABC is 45-degree, with 1 ≤ k ≤ K and -X ≤ a < b < c ≤ X.

For example, F(1, 10) = 41 and F(10, 100) = 12492. Find F(10⁶, 10⁹).

Written by gamwe6 who lives and works in San Francisco building useful things. You should follow him on Twitter