Counting the number of isosceles triangles in rectangular regular grids

In general graph theory, the only relationship between vertices are expressed via the edges. When the vertices are embedded in an Euclidean space, the geometric relationships between vertices and edges can be interesting objects of study. We look at the number of isosceles triangles where the vertices are points on a regular grid and show that they satisfy a recurrence relation when the grid is large enough. We also derive recurrence relations for the number of acute, obtuse and right isosceles triangles.
Comment: corrected minor typos