Heron triangles and a family of elliptic curves with rank zero

TL;DR

This paper explores the existence of rational-sided triangles with specific area and angle properties related to Heron triangles, and investigates their connection to elliptic curves and Diophantine equations.

Contribution

It introduces a novel investigation into triangles with rational sides and prescribed area and angle conditions linked to elliptic curves and number theory.

Findings

Existence conditions for triangles with area p and specific angle properties.

Connection between such triangles and solutions to particular Diophantine equations.

Analysis of elliptic curves with rank zero related to these geometric configurations.

Abstract

Given any positive integer $n$, it is well-known that there always exists a triangle with rational sides $a,b$ and $c$ such that the area of the triangle is $n$. For a given prime $p \not \equiv 1$ modulo $8$ such that $p^{2}+1=2q$ for a prime $q$, we look into the possibility of the existence of the triangles with rational sides with $p$ as the area and $\frac{1}{p}$ as $\tan \frac{\theta}{2}$ for one of the angles $\theta$. We also discuss the relation of such triangles with the solutions of certain Diophantine equations.