Let's classify all the possible equilateral triangles of interest. As there are more equilateral triangles in the proof, let's call those we want to count the counting triangles.
For each of the counting triangle asked, define its enclosing triangle be the minimal upright equilateral triangle that goes alone the network. Here, upright means the triangle has one side being horizontal, and the third vertex is above the horizontal side. Later in the text, the term upside-down will appear, meaning the opposite of upright, ie. one side being horizontal but the third vertex is below the horizontal side.
For example, here are some counting triangles (red) and their enclosing triangle (blue). The second counting triangle is upside-down.
We classify counting triangles by the side length of its enclosing triangle, as this directly related to how many these triangles are there in the original network. If the network has side N, we can find the number of ways
