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arXiv · 1809.09936

A unique pair of triangles

Yoshinosuke Hirakawa·Hideki Matsumura
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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.

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Yoshinosuke Hirakawa, Hideki Matsumura. 2018-09-26. A unique pair of triangles. https://doi.org/10.1016/j.jnt.2018.07.007

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