Ceva’s and Menelaus’ theorems in projective-metric spaces

  title={Ceva’s and Menelaus’ theorems in projective-metric spaces},
  author={{\'A}rp{\'a}d Kurusa},
  journal={Journal of Geometry},
  • Á. Kurusa
  • Published 12 July 2019
  • Mathematics
  • Journal of Geometry
AbstractWe prove that Ceva’s and Menelaus’ theorems are valid in a projective-metric space if and only if the space is any of the elliptic geometry, the hyperbolic geometry, or the Minkowski geometries.  

On Menelaus' and Ceva's theorems in Nil geometry

In this paper we deal with Nil geometry, which is one of the homogeneous Thurston 3-geometries. We define the “surface of a geodesic triangle” using generalized Apollonius surfaces. Moreover, we show

Classical Notions and Problems in Thurston Geometries

Of the Thurston geometries, those with constant curvature geometries (Euclidean E3, hyperbolic H3, spherical S3) have been extensively studied, but the other five geometries, H2×R, S2×R, Nil, S̃L2R,

Apollonius Surfaces, Circumscribed Spheres of Tetrahedra, Menelaus’s and Ceva’s Theorems in S2 × R and H2 × R Geometries

  • J. Szirmai
  • Mathematics
    The Quarterly Journal of Mathematics
  • 2021
In the present paper we study $\mathbf{S}^2\!\times\!\mathbf{R}$ and $\mathbf{H}^2\!\times\!\mathbf{R}$ geometries, which are homogeneous Thurston 3-geometries. We define and determine the



Ceva’s and Menelaus’ theorems characterize the hyperbolic geometry among Hilbert geometries

If a Hilbert geometry satisfies a rather weak version of either Ceva’s or Menelaus’ theorem for every triangle, then it is hyperbolic.

Incidence theorems in spaces of constant curvature

Certain analogs of the classic theorems of Menelaus and Ceva are considered for a hyperbolic surface, a sphere, and for three-dimensional hyperbolic and spherical spaces.

Hilbert’s fourth problem

Hilbert's fourth problem asks for the construction and the study of metrics on subsets of projective space for which the projective line segments are geodesics. Several solutions of the problem were

Support theorems for totally geodesic Radon transforms on constant curvature spaces

We prove a relation between the fc-dimensional totally geodesic Radon transforms on the various constant curvature spaces using the geodesic correspondence between the spaces. Then we use this

Hungary e-mail: kurusa@math.u-szeged