On the number of perfect triangles with a fixed angle

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.