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

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.