Triangle on Parabola

On the parabola $y = x^2/k$, three points $A(a, a^2/k)$, $B(b, b^2/k)$ and $C(c, c^2/k)$ 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 $1 \le k \le K$ and $-X \le a \lt b \lt c \le X$.

For example, $F(1, 10) = 41$ and $F(10, 100) = 12492$.
Find $F(10^6, 10^9)$.

This problem is not a straightforward one and coming up with an exact solution would require deep understanding of mathematical modelling and number theory. It would be a complex computation and would involve understanding the properties of quadruplets, Pythagoras theorem and geometric properties of parabola. Also, it is beyond the capability of an AI developed for conversation purposes to calculate such large inputs like $F(10^6, 10^9)$.

However, in a simpler form, the problem can be addressed if you want to know how to approach it:

1. Firstly, in order for a triangle to have a 45-degree angle, two of its sides must have the same length. We can use the distance formula to set up an equation for that.

2. Secondly, we should consider $a,b,c$ and $k$ to be in the domain of integers.

3. The problem can be reduced to finding the solutions to the following equations (considering the Pythagorean relation for a 45-degree triangle):

$(b-a)^2+(b^2-a^2)^2/k^2=(c-b)^2+(c^2-b^2)^2/k^2$

$(c-a)^2+(c^2-a^2)^2/k^2=(c-b)^2+(c^2-b^2)^2/k^2$

$(b-a)^2+(b^2-a^2)^2/k^2=(c-a)^2+(c^2-a^2)^2/k^2$

subject to the conditions provided in the problem: $1 \le k \le K$ and $-X \le a < b < c \le X$ 4. It is a quite challenging problem and it would require a careful case-by-case analysis or perhaps an ingenious insight to get the pattern of the solution. As a pointer though, the symmetry properties of the parabola and the problem's conditions might be exploited to simplify the problem. Please note that professional mathematicians or those experienced in competitive mathematics might be better suited for addressing this problem.

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