<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>In this paper we deal with <bold>Nil</bold> 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 trianglesistruein <bold>Nil</bold> space. In our work we will use the projective model of <bold>Nil</bold> geometry described by E. Molnár in [<xref ref-type="bibr" rid="j_ausm-2023-0008_ref_006">6</xref>].</p>
</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 trianglesistruein Nil space. In our work we will use the projective model of Nil geometry described by E. Molnár in [6].

On Menelaus’ and Ceva’s theorems in Nil geometry

Institute of Mathematics, Budapest University of Technology and Economics

Acta Universitatis Sapientiae, Mathematica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"On Menelaus\u2019 and Ceva\u2019s theorems in Nil geometry"}

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...

On Menelaus’ and Ceva’s theorems in Nil geometry

Published Online: Nov 15, 2023

Page range: 123 - 141

Received: Jul 11, 2022

DOI: https://doi.org/10.2478/ausm-2023-0008

Keywords
Thurston geometries, geometry, geodesic triangles, Menelaus’ and Ceva’s theorems

© 2023 Jenő Szirmai, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

On Menelaus’ and Ceva’s theorems in Nil geometry

Published Online: Nov 15, 2023

Page range: 123 - 141

Received: Jul 11, 2022

DOI: https://doi.org/10.2478/ausm-2023-0008

KeywordsThurston geometries, geometry, geodesic triangles, Menelaus’ and Ceva’s theorems

© 2023 Jenő Szirmai, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Thurston geometries, geometry, geodesic triangles, Menelaus’ and Ceva’s theorems