Napoleonic triangles on the sphere from similar chord triangles

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 $\beta \in (0,\pi \mathrm{/}2)$. For a non-equilateral $ABC$, we prove that the triangle $XYZ$ of spherical circumcentres is equilateral if and only if $\cos ({\theta }_a+{\theta }_b+{\theta }_c\pm \mathrm{\Omega }\mathrm{/}2)=0$, where ${\theta }_a,{\theta }_b,{\theta }_c$ are the base angles of the isosceles spherical triangles $XBC$, $YCA$, $ZAB$, $\mathrm{\Omega }$ 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 $\beta =\pi \mathrm{/}3$, and exhibits three families of solutions for $\beta >\pi \mathrm{/}3$, together with a fourth one when $\cos \beta <1\mathrm{/}4$.