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)∈ℚ, we obtain a two-parameter description of the solutions and, for fixed ρ, a family of curves Cρ which are generically of genus $2$. We determine the two singular specializations: there is no genuine pair for ρ=1/2, while ρ=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 Cρ(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.
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BibTeX
@misc{allen2026beyond,
title = {{Beyond the unique pair: a two-parameter family of rational triangles with equal perimeter and area}},
author = {Jesse Allen and Olivier Beaudry-Ogden and Théophane Bouchard and Matéo Frély and Matilde Lalín and Abel-Jimmy Oyono-Montoki and Berend Ringeling},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28020},
eprint = {2609.28020},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28020},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Beyond the unique pair: a two-parameter family of rational triangles with equal perimeter and area
AU - Jesse Allen
AU - Olivier Beaudry-Ogden
AU - Théophane Bouchard
AU - Matéo Frély
AU - Matilde Lalín
AU - Abel-Jimmy Oyono-Montoki
AU - Berend Ringeling
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28020
UR - https://arxiv.org/abs/2609.28020
AB - 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)∈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=1/2, while ρ=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(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.
ER -