# A unique pair of triangles

• Published in 2018
In the collections
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.

## Other information

key
Auniquepairoftriangles
type
article
2019-12-10
date_published
2018-09-14

### BibTeX entry

@article{Auniquepairoftriangles,
key = {Auniquepairoftriangles},
type = {article},
title = {A unique pair of triangles},
author = {Yoshinosuke Hirakawa and Hideki Matsumura},
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.},
comment = {},
}