# A unique pair of triangles

- Published in 2018
- Added on

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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.

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### 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} }