On rational triangles via algebraic curves

Abstract

A rational triangle is a triangle with rational side lengths. We consider three different families of rational triangles having a fixed side and whose vertices are rational points in the plane. We display a one-to-one correspondence between each family and the set of rational points of an algebraic curve. These algebraic curves are: a curve of genus 0, an elliptic curve and a genus~3 curve. We study the set of rational points on each of these curves and explicitly describe some of its rational points.