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.

arXiv comment5 pages, to appear in Journal of Number Theory, Some modifications are added to the article published online

References in corpus (1)

• Heron triangle and rhombus pairs with a common area and a common perimeter

Cited by in corpus (2)

• Amicable Heron triangles
• Some diophantine problems concerning a pair of rational triangles with a common circumradius