Year: 2026
Type: article
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].
Source: Results in Mathematics
Author: Jenő Szirmai
Institution: Budapest University of Technology and Economics
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4414896333
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