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883 - Remarkable Triangles

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Problem 883: Remarkable Triangles

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 $60^\circ$

diagram

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 $T(10^6)$.