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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.

##### MSC:
 51M04 Elementary problems in Euclidean geometries
##### Keywords:
$$n$$-simplex; Ceva’s theorem; Menelaus’ theorem
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