On rational triangles via algebraic curves

TL;DR

This paper explores families of rational triangles with a fixed side, establishing correspondences with algebraic curves of various genera and analyzing their rational points.

Contribution

It introduces a novel connection between rational triangles and algebraic curves of genus 0, elliptic, and genus 3, providing explicit descriptions of rational points.

Findings

Correspondence between rational triangles and algebraic curves

Explicit rational points on the curves of genus 0, elliptic, and genus 3

Insights into the structure of rational points on these 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 describe some of its rational points explicitly.