# 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.

### BibTeX entry

@article{Auniquepairoftriangles,
title = {A unique pair of triangles},
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.},
url = {http://arxiv.org/abs/1809.09936v1 http://arxiv.org/pdf/1809.09936v1},
year = 2018,
author = {Yoshinosuke Hirakawa and Hideki Matsumura},
comment = {},
urldate = {2019-12-10},
archivePrefix = {arXiv},
eprint = {1809.09936},
primaryClass = {math.NT},
collections = {easily-explained,fun-maths-facts,geometry}
}