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Ceva’s and Menelaus’ theorems for the \(n\)-dimensional space. (English) Zbl 0974.51016
Let \(A_1,\dots, A_{n+1}\) be an \(n\)-simplex in the \(n\)-dimensional Euclidean space \((n\geq 2)\) and \(B_{ij}\) be a point lying on the 1-dimensional space \(A_iA_j\), at which \(B_{ij}\neq A_i\), \(A_j(i,j\in \{1,\dots, n+ 1\}\), \(i\neq j)\). And let \(\Gamma_{ij}\) be the hyperplanes determined with the points from \(\{A_1,\dots,A_{n+1}\}\setminus\{A_i,A_j\}\) and the point \(B_{ij} \).
The author shows the following direct generalization of Ceva’s theorem respectively generalization of Menelaus’ theorem:
(i) The \({n+1 \choose 2}\) hyperplanes \(\Gamma_{\ell m} (\ell,m\in\{\ell,m \in\{1,\dots, n+1\})\), \(\ell\neq m)\) have a common point iff the following \({(n+1)!\over 3!}\) equalities are fulfilled: \({A_iB_{ij}\over B_{ij}A_j} \cdot{A_j B_{jk}\over B_{jk}A_k} \cdot {A_kB_{ik}\over B_{ik} A_i}=1\) \((i,j,k \in\{1, \dots,n+1\}\), \(i\neq j\neq k\neq i)\).
(ii) The \({n+1 \choose 2}\) points \(B_{lm}\) lie on one hyperplane iff the following \({(n+1)! \over 3!}\) conditions hold: \({A_iB_{ij}\over B_{ij} A_j} \cdot{A_jB_{jk} \over B_{jk}A_k} \cdot{A_kB_{ik} \over B_{ik}A_i}=-1\).
These theorems are valid already in the \(n\)-dimensional real affine space.

51M04 Elementary problems in Euclidean geometries
Full Text: EMIS EuDML