No perfect triangle is isosceles

Abstract

A perfect triangle is a triangle with rational sides, medians, and area. In this article, we use a similar strategy due to Pocklington to show that if ∆ is a perfect triangle, then it cannot be an isosceles triangle. It gives a partial answer to a question of Richard Guy, who asked whether any perfect triangles exist. No example has been found to date. It is widely believed that such a triangle does not exist.Theorem 1.1. Given 0 < θ < π where θ = π 2 , there are, up to similarity, finitely many Heron triangles with an angle θ and with two rational medians.