Hyperbolic Heron Triangles and Elliptic Curves

TL;DR

This paper introduces hyperbolic Heron triangles, parametrizes them via elliptic curves, and proves the existence of infinitely many such triangles with given angles and areas, extending classical rational triangle problems into hyperbolic geometry.

Contribution

It defines hyperbolic Heron triangles, provides parametrizations using elliptic curves, and demonstrates the infinite existence of such triangles with specified angles and areas.

Findings

Infinitely many hyperbolic Heron triangles exist for any admissible angle and area.

Hyperbolic Heron triangles can be parametrized by rational points on elliptic curves.

The hyperbolic congruent number problem has infinitely many solutions.

Abstract

We define hyperbolic Heron triangles (hyperbolic triangles with "rational" side-lengths and area) and parametrize them in two ways as rational points of certain elliptic curves. We show that there are infinitely many hyperbolic Heron triangles with one angle $\alpha$ and area $A$ for any (admissible) choice of $\alpha$ and $A$; in particular, the congruent number problem has always infinitely many solutions in the hyperbolic setting. We also explore the question of hyperbolic triangles with a rational median and a rational area bisector (median splitting the triangle in half).