Menelaus' and Ceva's theorems are formulated and proved for translation triangles in Nil, Sol, and ~SL2R spaces using geometry-specific simple ratios.
Translation-like Apollonius and triangular surfaces in non-constant curvature Thurston geometries
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In the present paper we deal with non-constant curvature Thurston geometries \cite{M97}, \cite{S}, \cite{Sz22-3},\cite{W06}. We define and determine the generalized trans\-lation-like Apollonius surfaces and thus also bisector surfaces as a special case. Moreover, we give a possible definition of the "surface of a translation-like triangle" in each investigated geometry. In our work we will use the projective model of Thurston geometries described by E. Moln\'ar in \cite{M97}.
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Menelaus' and Ceva's theorems for translation triangles in Thurston geometries
Menelaus' and Ceva's theorems are formulated and proved for translation triangles in Nil, Sol, and ~SL2R spaces using geometry-specific simple ratios.