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

2025/06/02 by Szirmai, Jenő · 1 citation
#52C35 #53A20 #53A35 #53B20 #FOS: Mathematics #Geometric Topology (math.GT) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2506.01354

Abstract

After having investigated and defined the ``surface of a translation-like triangle" in each non-constant curvature Thurston geometry \citeCs-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. Molnár in \citeM97.

Citations

• Translation-like Apollonius and triangular surfaces in non-constant curvature Thurston geometries
• Interior angle sums of geodesic triangles and translation-like isoptic surfaces in Sol geometry
• Classical Notions and Problems in Thurston Geometries
• On Menelaus' and Ceva's theorems in Nil geometry
• Lattice coverings by congruent translation balls using translation-like bisector surfaces in Nil geometry
• Bisector surfaces and circumscribed spheres of tetrahedra derived by translation curves in \SOL geometry
• Triangle angle sums related to translation curves in \SOL geometry
• NIL geodesic triangles and their interior angle sums
• On hyperbolic analogues of some classical theorems in spherical geometry
• A candidate to the densest packing with equal balls in the Thurston\n geometries
• Classification of Sol lattices
• The Geometries of 3-Manifolds
• Curvatures of left invariant metrics on lie groups

Cited by

• Lattice-like Packings and Coverings with Congruent Translation Balls and Cylinders in Sol geometry

Related

• Translation-like Apollonius and triangular surfaces in non-constant curvature Thurston geometries
• On Menelaus' and Ceva's theorems in Nil geometry
• Translation-like isoptic surfaces and angle sums of translation triangles in \NIL geometry
• Classical Notions and Problems in Thurston Geometries
• Bisector surfaces and circumscribed spheres of tetrahedra derived by translation curves in \SOL geometry