Nearequilateral

Nearequilateral is a term used informally in geometry to describe a triangle whose side lengths or angles are close to those of an equilateral triangle. There is no formal standard definition, but a nearequilateral triangle is typically defined as one where the three side lengths are nearly equal or equivalently where each interior angle deviates only slightly from 60 degrees.

Definition and measures:

Let a, b, c be the side lengths of a triangle. The triangle is considered nearequilateral if max(a, b, c) / min(a, b, c) is close to 1, or if the angular deviations δ_A = |A − 60°|, δ_B = |B − 60°|, δ_C = |C − 60°| are small. The equilateral triangle (all sides equal and all angles 60°) is the exact case of a nearequilateral triangle.

Construction and variability:

A nearequilateral triangle can be obtained by perturbing a regular (equilateral) triangle slightly, for example by lengthening one side a small amount or slightly changing one angle. These perturbations keep the figure close to symmetry, with all elements remaining nearly proportional.

Applications:

Near-equilateral triangles appear in mesh generation and numerical methods because elements that are nearly equilateral tend to reduce interpolation error and improve stability. They are also studied in problems where an exact equilateral triangle cannot be achieved due to constraints, yet a close approximation is sufficient.

See also:

Equilateral triangle, Isosceles triangle, Scalene triangle, Mesh quality, Triangulation.