Beyond the unique pair: a two-parameter family of rational triangles with equal perimeter and area

Abstract

We study pairs consisting of a rational isosceles triangle and a rational triangle having a prescribed angle $θ$, with equal perimeter and equal area. Writing $ρ=\cos(θ)\in\mathbb{Q}$, we obtain a two-parameter description of the solutions and, for fixed $ρ$, a family of curves $C_ρ$ which are generically of genus $2$. We determine the two singular specializations: there is no genuine pair for $ρ=\frac{1}{2}$, while $ρ=\frac{47}{49}$ admits infinitely many genuine pairs. We completely parametrize two natural one-parameter families in which the $θ$-triangle is isosceles, and prove several density results for values of $ρ$ admitting one or more genuine pairs. For thirty-five explicit values of $ρ$, Chabauty--Coleman computations determine $C_ρ(\mathbb{Q})$ completely and show that the corresponding pair is unique up to homothety. We also study multiplicity when one triangle is fixed; the condition for a scalene $θ$-triangle to have two isosceles partners leads to an elliptic $K3$ surface and yields a dense set of such values of $ρ$. Finally, we construct infinite families with at least four genuine pairs and exhibit examples carrying at least five, and in two cases at least six, genuine pairs.