# 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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## Other information

- key
- Auniquepairoftriangles
- type
- article
- date_added
- 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 = {}, date_added = {2019-12-10}, date_published = {2018-09-14}, urls = {http://arxiv.org/abs/1809.09936v1,http://arxiv.org/pdf/1809.09936v1}, collections = {Easily explained,Fun maths facts,Geometry}, url = {http://arxiv.org/abs/1809.09936v1 http://arxiv.org/pdf/1809.09936v1}, year = 2018, urldate = {2019-12-10}, archivePrefix = {arXiv}, eprint = {1809.09936}, primaryClass = {math.NT} }