An Infinite Family of Primitive Heron Triangles with Two Sides as Perfect Squares

Abstract

A primitive Heron triangle is a triangle with integral sides and integral area where the greatest common divisor of the lengths of its sides is $1$. By utilizing the theory of elliptic curves, we prove that there exist infinitely many primitive Heron triangles with two sides being perfect squares. In this process, we nest one elliptic curve into another and find a surprising rational point. All the Heron triangles corresponding to this rational point are primitive. This result would imply the possible existence of infinitely many primitive Heron triangles with all three sides being perfect squares.