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.
