Menelaus’ and Ceva’s Theorems for Translation Triangles in Thurston Geometries

Abstract After having analysed and defined the “surface of a translation-like triangle" in each non-constant curvature Thurston geometry [7], i.e. in $$\textbf{S}^2\!\times \!\textbf{R}, \textbf{H}^2\!\times \!\textbf{R}, \widetilde{\textbf{S}\textbf{L}_2\textbf{R}}, \textbf{Nil}, \textbf{Sol}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mspace/> <mml:mo>×</mml:mo> <mml:mspace/> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mspace/> <mml:mo>×</mml:mo> <mml:mspace/> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mover> <mml:mrow> <mml:mi>S</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mi>R</mml:mi> </mml:mrow> <mml:mo>~</mml:mo> </mml:mover> <mml:mo>,</mml:mo> <mml:mi>Nil</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Sol</mml:mi> </mml:mrow> </mml:math> , 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].