Menelaus’ and Ceva’s Theorems for Translation Triangles in Thurston Geometries
Abstract
Abstract After having analysed and defined the “surface of a translation-like triangle" in each non-constant curvature Thurston geometry [7], i.e. in $$\textbf{S}^2\!\times \!\textbf{R}, \textbf{H}^2\!\times \!\textbf{R}, \widetilde{\textbf{S}\textbf{L}_2\textbf{R}}, \textbf{Nil}, \textbf{Sol}$$ S 2 × R , H 2 × R , S L 2 R ~ , Nil , Sol , we extend the famous Menelaus and Ceva theorems for translation triangles in the mentioned spaces by generalizing the concept of simple ratio. Our method makes possible to transfer further classical Euclidean theorems and notions to the above Thurston geometries. In our work we will use the projective models of Thurston geometries introduced by Molnár in [10].
Fields of Study & Subjects
Robotic Mechanisms and Dynamics
Mathematics and Applications
Mathematical Dynamics and Fractals
Publication details
- Journal / source
- Results in Mathematics
- Publisher
- Springer Science and Business Media LLC
- Volume
- 81
- Issue
- 6
- Publication date
- 3 August 2026
- Language
- EN