No perfect triangle is isosceles

TL;DR

This paper proves that perfect triangles, with rational sides, medians, and area, cannot be isosceles, providing partial evidence towards the conjecture that no perfect triangles exist at all.

Contribution

It introduces a novel application of Pocklington's strategy to demonstrate the non-existence of isosceles perfect triangles.

Findings

Perfect triangles cannot be isosceles.

Supports the conjecture that perfect triangles may not exist.

Provides a partial answer to Richard Guy's question.

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 $\Delta$ 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.