On Moving Chords in Constant Curvature 2-Manifolds

An introduction to non-Euclidean geometry is given and a new approach to prove a generalization of Holditch's theorem in 2-dimensional constant curvature manifolds is presented. Moreover, the same procedure also makes possible to obtain generalized versions of Barbier's theorem for constant width curves and of Steiner's formulae for parallel curves. The main fact is that a general formula relating the geodesic curvatures of the involved curves is found, in such a way these results can be derived from there. Regularity of some curves generated geodesically from another is also studied.