Obtuse Angled Triangles

Consider the set $S(r)$ of points $(x,y)$ with integer coordinates satisfying $|x| + |y| \le r$.
Let $O$ be the point $(0,0)$ and $C$ the point $(r/4,r/4)$.
Let $N(r)$ be the number of points $B$ in $S(r)$, so that the triangle $OBC$ has an obtuse angle, i.e. the largest angle $\alpha$ satisfies $90^\circ \lt \alpha \lt 180^\circ$.
So, for example, $N(4)=24$ and $N(8)=100$.

What is $N(1\,000\,000\,000)$?

This problem may be considered as a geometry problem on the coordinate plane. To start, let’s analyze the nature of the triangle OBC and the obtuse angle.

An angle in a triangle is obtuse if and only if it’s opposite a side that is the longest side in the triangle. Therefore, we are considering points B such that either OB or CB is the longest side of triangle OBC.

If OB is the longest side, this means that B must lie on the line OC and outside of the segment OC. As OC lies on the line y=x, we look at points B such that y=x and x>r/4. This clearly gives r/4 of these points.

If CB is the longest side, then the points B must lie inside the square formed by (-r/2,-r/2), (r/2,r/2), (-r/2, r/2), and (r/2, -r/2) but outside the circle with radius r/2 centered at O. The number of lattice points within this square is given by $(r+1)^2$.

The number of lattice points within the circle can be estimated by the formula for the area of a circle, as the density of lattice points is roughly 1 per unit square. So, this estimated number of beats is $\pi (r/2)^2 = \pi r^2/4$. Therefore, the estimated number of points in this annulus is $(r+1)^2 – \pi r^2 / 4$.

However, this answer isn’t precise because the density of lattice points isn’t exactly 1 per unit square throughout the plane (it actually depends on the rationals of the radius r). Nevertheless, it’s reasonable to assume that this estimate provides the leading term in the actual result.

For r = 1,000,000,000, we then have $N(1\,000\,000\,000) \approx r/4 + (r+1)^2 – \pi r^2 / 4$.

This is a rough calculation method. For a substantial proof, more rigorous geometric and number theoretic methods would be necessary. A concise answer might not be possible immediately without computational assistance.

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