On the number of perfect triangles with a fixed angle

TL;DR

This paper proves that for any fixed angle, only finitely many perfect triangles with rational sides, medians, and area exist, and that no algebraic curve over the reals contains an infinite rational median set.

Contribution

It establishes finiteness results for perfect triangles with a fixed angle and shows that infinite rational median sets cannot lie on irreducible algebraic curves.

Findings

Finiteness of perfect triangles with a fixed angle

No infinite rational median set on irreducible algebraic curves

Application of rational distance set techniques to triangle problems

Abstract

Richard Guy asked the following question: can we find a triangle with rational sides, medians, and area? Such a triangle is called a \emph{perfect triangle} and no example has been found to date. It is widely believed that such a triangle does not exist. Here we use the setup of Solymosi and de Zeeuw about rational distance sets contained in an algebraic curve, to show that for any angle $0<\theta < \pi$, the number of perfect triangles with an angle $\theta$ is finite. A \emph{rational median set} $S$ is a set of points in the plane such that for every three non collinear points $p_1,p_2,p_3$ in $S$ all medians of the triangle with vertices at $p_i$'s have rational length. The second result of this paper is that no irreducible algebraic curve defined over $\mathbb{R}$ contains an infinite rational median set.