Menelaus' and Ceva's theorems for translation triangles in Thurston geometries

TL;DR

This paper generalizes Menelaus' and Ceva's theorems for translation triangles within various Thurston geometries, enabling the transfer of classical Euclidean theorems to non-constant curvature spaces.

Contribution

It extends classical triangle theorems to Thurston geometries using projective models, broadening their applicability beyond Euclidean space.

Findings

Generalization of Menelaus' theorem in Thurston geometries

Generalization of Ceva's theorem in Thurston geometries

Method for transferring Euclidean theorems to non-constant curvature spaces

Abstract

After having investigated and defined the ``surface of a translation-like triangle" in each non-constant curvature Thurston geometry \cite{Cs-Sz25}, we generalize the famous Menelaus' and Ceva's theorems for translation triangles in the mentioned spaces. The described method makes it possible to transfer further classical Euclidean theorems and notions to Thurston geometries with non-constant curvature. In our work we will use the projective models of Thurston geometries described by E. Moln\'ar in \cite{M97}.