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 $60^{\circ}$

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 $\leq r$ . 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 $T (10^{6})$ .

卓越三角形

本题所考虑的三角形的顶点均在六边形格阵上。六边形格阵是平面上的一种格点阵，其中每个格点周围都有六个相邻的、均匀分布的格点，其距离均为 $1$ 。

所谓卓越三角形是满足以下条件的三角形：

其三个顶点及其内心都落在格点上
至少一个角为 $60^{\circ}$

如上所示是四个卓越三角形，其 $60^{\circ}$ 的角被标记为红色。三角形A和B的内接圆半径为 $1$ ，C的内接圆半径为 $\sqrt{3}$ ，而D的内接圆半径为 $2$ 。

记 $T (r)$ 为所有内接圆半径小于等于 $r$ 的卓越三角形的数目。旋转和翻转形成的不同三角形，如上图所示的三角形A和B，在计数时分别计算；而直接平移则不重复计算，也就是说，在格阵不同的位置绘制的完全相同的三角形只计算一次。

已知 $T (0.5) = 2$ ， $T (2) = 44$ ， $T (10) = 1302$ 。

求 $T (10^{6})$ 。