A unique pair of triangles
arXiv:1809.09936 · 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 -adic abelian integrals.
5 pages, to appear in Journal of Number Theory, Some modifications are added to the article published online