A unique pair of triangles

2018/09/26 by Yoshinosuke Hirakawa, Hideki Matsumura · 1 voice
#math.NT

paper · pdf · doi:10.1016/j.jnt.2018.07.007

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.

Discussions

• of integer-sided triangles—one right and one isosceles—with the same area and the same perimeter (up to similarity)! Those are the (377, 135, 352) and (366, 366, 132) triangles. The proof of 1 is not [bsky, 6 points, 0 comments]

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