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 $β\in(0,π/2)$. For a non-equilateral $ABC$, we prove that the triangle $XYZ$ of spherical circumcentres is equilateral if and only if $\cos(θ_a+θ_b+θ_c\pmΩ/2)=0$, 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 $\cosβ<1/4$.