Integer triangles with a rational ratio of circumcircle radius to excircle radius

TL;DR

This paper investigates integer triangles with a rational ratio of circumcircle to excircle radii, linking the problem to elliptic curves and establishing conditions for the existence of such triangles.

Contribution

It establishes a connection between the ratio R/r and elliptic curves, providing criteria for the existence and infinitude of integer triangles with rational R/r.

Findings

For general triangles, R/r > 1/4.

Existence of rational triangles depends on rational points on elliptic curves.

Infinite solutions exist when the elliptic curve has positive rank.

Abstract

We consider the problem of finding integer triangles with $R/r$ a positive rational, where $R$ and $r$ are the radii of the circumcircle and an excircle, respectively. We show that for general triangles $R/r>1/4$ applies. The equation $R/r=N$ turns out to be related to the elliptic curve $\mathcal{E}_N$ given by $v^2=u^3+2(2N^2+2N-1)u^2-(4N-1)u$. If $N>1/4$ is rational, then the torsion group of $\mathcal{E}_N$ is $\mathbb Z/2\mathbb Z\times\mathbb Z/6\mathbb Z$ if $N(N+2)$ is a square and $\mathbb Z/6\mathbb Z$ otherwise. We show that a rational triangle with rational ratio $R/r=N$ exists if and only if $N>1/4$ and there exists a rational non-torsion point on the curve $\mathcal{E}_N$ which satisfies a certain condition. Furthermore, we show that the rank of $\mathcal{E}_N$ is positive when $N = m^2 \pm 1>1/4$ for a rational $m$. We also show that on every curve $\mathcal{E}_N$ whose rank is positive, there are infinitely many rational points which lead to infinitely many non-similar integer triangles with $R/r=N$.