Counting the number of isosceles triangles in rectangular regular grids

TL;DR

This paper investigates the enumeration of isosceles triangles formed by grid points, deriving recurrence relations for their counts, including special types like acute, obtuse, and right triangles, in large regular grids.

Contribution

It introduces recurrence relations for counting various types of isosceles triangles in large regular grids, a novel approach in geometric combinatorics.

Findings

Recurrence relations for isosceles triangles in grids

Formulas for acute, obtuse, and right isosceles triangles

Applicable to large grid sizes

Abstract

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.