Hyperbolic classification for isosceles triangles with prescribed apex angle

Determine the classification in the hyperbolic plane of triangles for which erecting isosceles hyperbolic triangles with prescribed apex angle $2\beta$ produces a hyperbolic analogue of the spherical Napoleonic configuration, taking into account that the Euclidean chord-angle definition of a $\beta$-point has no direct intrinsic hyperbolic analogue.

Background

The paper studies a spherical construction based on isosceles chord triangles and classifies the spherical input triangles for which the three resulting circumcentres form an equilateral triangle. It notes that the original Euclidean chord-angle definition of a $\beta$-point does not transfer directly to the hyperbolic plane. A suggested hyperbolic replacement is to erect isosceles hyperbolic triangles with apex angle $2\beta$; their base angles are automatically smaller than $\pi/2-\beta$, removing the acute-versus-obtuse ambiguity present in the spherical setting. The corresponding classification, including comparison with the known $\beta=\pi/3$ case, is explicitly left unresolved.