Scholarlink

<p>After having analysed and defined the “surface of a translation-like triangle" in each non-constant curvature Thurston geometry [<CitationRef CitationID="CR7">7</CitationRef>], i.e. in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textbf{S}^2\!\times \!\textbf{R}, \textbf{H}^2\!\times \!\textbf{R}, \widetilde{\textbf{S}\textbf{L}_2\textbf{R}}, \textbf{Nil}, \textbf{Sol}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="bold">S</mi> <mn>2</mn> </msup> <mspace width="-0.166667em" /> <mo>×</mo> <mspace width="-0.166667em" /> <mi mathvariant="bold">R</mi> <mo>,</mo> <msup> <mi mathvariant="bold">H</mi> <mn>2</mn> </msup> <mspace width="-0.166667em" /> <mo>×</mo> <mspace width="-0.166667em" /> <mi mathvariant="bold">R</mi> <mo>,</mo> <mover accent="true"> <mrow> <mi mathvariant="bold">S</mi> <msub> <mi mathvariant="bold">L</mi> <mn>2</mn> </msub> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">~</mo> </mover> <mo>,</mo> <mi mathvariant="bold">Nil</mi> <mo>,</mo> <mi mathvariant="bold">Sol</mi> </mrow> </math></EquationSource> </InlineEquation>, 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 [<CitationRef CitationID="CR10">10</CitationRef>].</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

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

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2026年8月3日

查看全文
  • Jenő Szirmai

摘要

After having analysed and defined the “surface of a translation-like triangle" in each non-constant curvature Thurston geometry [7], i.e. in \(\textbf{S}^2\!\times \!\textbf{R}, \textbf{H}^2\!\times \!\textbf{R}, \widetilde{\textbf{S}\textbf{L}_2\textbf{R}}, \textbf{Nil}, \textbf{Sol}\) S 2 × R , H 2 × R , S L 2 R ~ , Nil , Sol , 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].