A unique pair of triangles

TL;DR

This paper proves the unique existence of a pair consisting of a rational right triangle and a rational isosceles triangle sharing the same perimeter and area, using advanced techniques in algebraic geometry.

Contribution

It establishes the uniqueness of such a pair by analyzing rational points on a hyperelliptic curve through 2-descent and p-adic integrals.

Findings

Existence of a unique pair of rational triangles with shared perimeter and area.

Application of hyperelliptic curve rational point analysis to geometric problems.

Use of advanced algebraic geometry methods in triangle classification.

Abstract

A rational triangle is a triangle with sides of rational lengths. In this short note, we prove that there exists a unique pair of a rational right triangle and a rational isosceles triangle which have the same perimeter and the same area. In the proof, we determine the set of rational points on a certain hyperelliptic curve by a standard but sophisticated argument which is based on the 2-descent on its Jacobian variety and Coleman's theory of $p$-adic abelian integrals.