Synthetic proof of the angle criterion

Develop a synthetic proof of the angle criterion for the spherical Napoleonic construction with similar chord triangles, replacing the quaternion-based proof obtained by composing rotations about the constructed centres.

Background

The paper proves its principal angle criterion by composing rotations about the points X, Y, and Z and evaluating the resulting composition with quaternions. The criterion characterizes when the triangle of spherical circumcentres is equilateral in terms of the vanishing of a cosine involving the base-angle sum and the spherical area. The author explicitly asks whether this algebraic and rotational argument can be replaced by a synthetic geometric proof.

References

The appendix proves the angle criterion by composing the rotations about $X,Y,Z$, and uses quaternions to compute the composition. Can this argument be replaced by a synthetic proof?

— Napoleonic triangles on the sphere from similar chord triangles  (2610.21408 - Tran, 15 Oct 2026) in Section 6, Further questions, first Question

For which $ABC$ and $\beta$ is $DEF$ equilateral? Proposition~\ref{prop:DEF} gives a sufficient condition, but Remark~\ref{rem:DEF}(2) shows that the answer extends beyond inputs with equilateral $XYZ$.

— Napoleonic triangles on the sphere from similar chord triangles  (2610.21408 - Tran, 15 Oct 2026) in Section 6, Further questions, third Question

The Petr--Douglas--Neumann theorem extends Napoleon's theorem to polygons (see the references in ). Is there a spherical counterpart obtained by imposing similarity on chord polygons?

— Napoleonic triangles on the sphere from similar chord triangles  (2610.21408 - Tran, 15 Oct 2026) in Section 6, Further questions, fourth Question