Pairs of rational triangles with equal symmetric invariants

Abstract

The semiperimeter s, inradius r and circumradius R are the symmetric invariants of a triangle. In this short note, we complement previous work of Hirakawa and Matsumura by determining all pairs (up to similitude) consisting of a rational right angled triangle and a rational isosceles triangle having two corresponding symmetric invariants equal. In the proof of our main theorems, we determine the set of rational points on two hyperelliptic curves of genus 3. The first of these can be settled via a classical descent argument, whereas the second one is proved using ‘‘elliptic curve Chabauty’’.