I have very little experience in non-euclidean geometries, but I was wondering if there was some equivalent of the Euler line for some triangle in non-euclidean geometries.
Could we define a "right angle" to mean "an angle that is congruent to its supplementary angle", use geodesics instead of lines, to get the circumcenter, centroid, and orthocenter, and show there exists some geodesic that passes through all three points?
Thanks!