Spherical Heron triangles and elliptic curves

TL;DR

This paper introduces spherical Heron triangles with rational sides and angles, parametrizes them using elliptic curves, and explores their properties, including solutions to the congruent number problem and special median configurations.

Contribution

It provides a new framework for spherical Heron triangles via elliptic curves and demonstrates the existence of infinitely many solutions for certain areas.

Findings

Infinitely many solutions to the spherical congruent number problem for most areas.

Existence of a spherical Heron triangle with rational medians.

Analysis of triangles with a single rational median or area bisector.

Abstract

We define spherical Heron triangles (spherical triangles with "rational" side-lengths and angles) and parametrize them via rational points of certain families of elliptic curves. We show that the congruent number problem has infinitely many solutions for most areas in the spherical setting and we find a spherical Heron triangle with rational medians. We also explore the question of spherical triangles with a single rational median or a single a rational area bisector (median splitting the triangle in half), and discuss various problems involving isosceles spherical triangles.