In this problem we consider triangles drawn on a hexagonal lattice, where each lattice point in the plane has six neighbouring points equally spaced around it, all distance $1$ away.
We call a triangle remarkable if
- All three vertices and its incentre lie on lattice points
- At least one of its angles is $60^\circ$
Above are four examples of remarkable triangles, with $60^\circ$ angles illustrated in red. Triangles A and B have inradius $1$; C has inradius $\sqrt{3}$; D has inradius $2$.
Define $T(r)$ to be the number of remarkable triangles with inradius $\le r$. Rotations and reflections, such as triangles A and B above, are counted separately; however direct translations are not. That is, the same triangle drawn in different positions of the lattice is only counted once.
You are given $T(0.5)=2$, $T(2)=44$, and $T(10)=1302$.
Find $T(10^6)$.
solutions/s883.md.