Napoleonic triangles on the sphere from similar chord triangles
Abstract
We study a variant of Napoleon's construction on the unit sphere, in which the triangles erected on the sides of a spherical triangle ABC are required to have similar isosceles chord triangles with apex angle β∈(0,π/2). For a non-equilateral ABC, we prove that the triangle XYZ of spherical circumcentres is equilateral if and only if cos(θa+θb+θc±Ω/2)=0, where θa,θb,θc are the base angles of the isosceles spherical triangles XBC, YCA, ZAB, Ω is the area of ABC, and the sign + (resp.~−) corresponds to the outward (resp.~inward) construction. We derive from this an explicit classification in terms of two quadrics and a cubic inequality. It recovers the theorem of Dipierro, Noakes and Valdinoci for β=π/3, and exhibits three families of solutions for β>π/3, together with a fourth one when cosβ<1/4.
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BibTeX
@misc{tran2026napoleonic,
title = {{Napoleonic triangles on the sphere from similar chord triangles}},
author = {Quang Hung Tran},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21408},
eprint = {2610.21408},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21408},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Napoleonic triangles on the sphere from similar chord triangles
AU - Quang Hung Tran
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21408
UR - https://arxiv.org/abs/2610.21408
AB - We study a variant of Napoleon's construction on the unit sphere, in which the triangles erected on the sides of a spherical triangle ABC are required to have similar isosceles chord triangles with apex angle β∈(0,π/2). For a non-equilateral ABC, we prove that the triangle XYZ of spherical circumcentres is equilateral if and only if cos(θ_a+θ_b+θ_c±Ω/2)=0, where θ_a,θ_b,θ_c are the base angles of the isosceles spherical triangles XBC, YCA, ZAB, Ω is the area of ABC, and the sign + (resp.~−) corresponds to the outward (resp.~inward) construction. We derive from this an explicit classification in terms of two quadrics and a cubic inequality. It recovers the theorem of Dipierro, Noakes and Valdinoci for β=π/3, and exhibits three families of solutions for β>π/3, together with a fourth one when cosβ<1/4.
ER -