# Incidence theorems in spaces of constant curvature

@article{Masaltsev1994IncidenceTI,
title={Incidence theorems in spaces of constant curvature},
author={L. A. Masal’tsev},
journal={Journal of Mathematical Sciences},
year={1994},
volume={72},
pages={3201-3206}
}
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.
10 Citations
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
• Mathematics
• 2011
In this study, we give isogonal conjugates from major contributions of the modern synthetic geometry of the hyperbolic triangle in Poincaré upper half plane model of hyperbolic geometry.
• Mathematics
• 1997
The purpose of this paper is to state and prove a theorem (the CMS Theorem) which generalizes the familiar Ceva's Theorem and Menelaus' Theorem of elementary Euclidean geometry. The theorem concernsn
• Mathematics
Am. Math. Mon.
• 2017
Another proof of the Steiner—Routh theorem for tetrahedra is given, where methods of elementary geometry are combined with the inclusion—exclusion principle, and this approach is generalized to (n — 1)-dimensional simplices.
• A. Glutsyuk
• Mathematics
Journal of Fixed Point Theory and Applications
• 2022
Reflection in planar billiard acts on oriented lines. For a given closed convex planar curve $$\gamma$$ γ , the string construction yields a one-parameter family $$\Gamma _p$$ Γ p of nested billiard
• M. Musielak
• Mathematics
Ukrainian Mathematical Journal
• 2021
We prove the following theorems: (1) every spherical convex body W of constant width ΔW≥π2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}
• A. Glutsyuk
• Materials Science
Journal of Fixed Point Theory and Applications
• 2022
Reflection in planar billiard acts on oriented lines. For a given closed convex planar curve γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}
We prove the following theorems: (1) every spherical convex body W of constant width Δ W ≥ π 2 $$\varDelta (W)\ge \frac{\uppi}{2}$$ can be covered by a disk of radius Δ W + arcsin 2 3 3 cos Δ W 2 −
It is proved under reasonable hypotheses that the number of determining modes is bounded by $c\mathcal{G}^{1/2}+ \epsilon^{1-2}M$, where $1/\ep silon$ is the rotation rate and $M$ depends on up to third derivatives of the forcing.
In this study, we give isogonal conjugates from major contributions of the modern synthetic geometry of the triangle in hyperbolic plane. Key Words: Hyperbolic Ceva theorem and Hyperolic sines theorem