In euclidean geometry, the locus of points from which a given segment AB subtends a given angle of alpha is made up of two arcs (https://pt.wikipedia.org/wiki/Par_de_arcos_capazes). My question is what are these loci in non euclidean geometries (spherical, taxicab, Poincaré hyperbolic)? In portuguese these loci are called "arco capaz". ChatGPT says that in english it's called apollonius circle. is this correct?

$\begingroup$ You haven't mentioned to ChatGPT any interest in nonEuclidean geometries, so why would it answer accordingly? At least as far as Wiki knows, the term "Apollonius circle" is used only in Euclidean geometry (although it does not seem to be what you are seeking even there). $\endgroup$– LSpiceCommented Apr 9, 2023 at 21:05
1 Answer
These loci may be called the isoptic curves of the segment. In the spherical and hyperbolic geometry, the set of points from which a segment is seen under the right angle (the orthoptic curve of the segment) is a spherical resp. hyperbolic conic. More generally, the orthoptic curve of any spherical/hyperbolic conic is a spherical/hyperbolic conic.
If the angle is different from $90^\circ$, then the isoptic of a segment seems to be a curve of degree four in the projective model, see the 2014 article "Isoptic curves of conic sections in constant curvature geometries" by Csima and Szirmai in "Mathematical communications".