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 $T(10^6)$.
