On Menelaus' and Ceva's theorems in Nil geometry

TL;DR

This paper explores classical geometric theorems Menelaus' and Ceva's within Nil geometry, demonstrating their validity on geodesic triangles using generalized surfaces and projective models.

Contribution

It extends fundamental Euclidean theorems to Nil geometry, providing new insights into geodesic triangles and their properties in this Thurston 3-geometry.

Findings

Menelaus' condition applies to lines on geodesic triangle surfaces in Nil geometry.

Ceva's theorem holds true for geodesic triangles in Nil space.

Introduction of generalized Apollonius surfaces in Nil geometry.

Abstract

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 that the "lines" on the surface of a geodesic triangle can be defined by the famous Menelaus' condition and prove that Ceva's theorem for geodesic triangles is true in $\NIL$ space. In our work we will use the projective model of $\NIL$ geometry described by E. Moln\'ar in \cite{M97}.