原論文貼っとく

A unique pair of triangles
Yoshinosuke Hirakawa, Hideki Matsumura

Journal of Number Theory
In Press, Corrected Proof, Available online 24 August 2018

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.

MSC
primary14G05secondary11G3011Y50
Keywords
Diophantine geometryHyperelliptic curvesRational triangles

https://www.sciencedirect.com/science/article/pii/S0022314X18302269