A Complete Characterization of Heron Triangles with Two Perfect Square Sides and the All-Square Equivalence Condition

TL;DR

This paper characterizes Heron triangles with perfect square sides, showing finitely many with three such sides and infinitely many with two, through algebraic curve analysis and elliptic curves.

Contribution

It provides a complete characterization of Heron triangles with two or three perfect square sides using algebraic and elliptic curve methods.

Findings

Finitely many Heron triangles have three perfect square sides.

Infinitely many primitive Heron triangles have two perfect square sides.

Complete classification achieved for triangles with two perfect square sides.

Abstract

A Heron triangle is a triangle whose side lengths and area are all positive integers. If the greatest common divisor of the three side lengths is $1$, it is called a primitive Heron triangle. In this paper, we give an equivalent condition for Heron triangles with all three sides being perfect squares, which reduces to finding non-trivial rational points on a family of algebraic curves of genus $3$. This leads us to believe that only finitely many Heron triangles with three perfect square sides exist. Using a specific elliptic curve, we completely characterize all Heron triangles with two sides that are perfect squares, and obtain a family of parametric solutions that yield primitive Heron triangles. This implies that there are infinitely many primitive Heron triangles having two sides as perfect squares.