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Orthocentric simplices as the true generalizations of triangles. (English) Zbl 1287.51011

The main purpose of this paper is to explore the differences and analogies of the Euclidean spaces \(\mathbb{R}^{d}\), \(d\geq 3\), focusing on the fact that tetrahedra and simplices in higher dimension do not necessarily have orthocenters. This lack of orthocenter leads, on one hand, to the consideration of orthocentric simplices and, on the other hand, to the attempt of replacement of the orthocenter by other points like the Monge point or the generalized Gergonne point.
There are some theorems about triangles that refer to the orthocenter, like the Euler line theorem and the nine-point circle theorem, that unexpectedly, when generalized to dimension 3, are satisfied not only by orthocentric tetrahedra but by all the tetrahedra. However, there are also triangle theorems that don’t make reference to the orthocenter, like the pons asinorum theorem, the open mouth theorem, the theorems partaining to Hilbert’s third problem, the purity of corner angles, and the degrees of regularity implied by coincidences of certain pair of centers, that, when are extended to higher dimension, are true only for orthocentric simplices. This behavior of simplices brings us to the discussion if the orthocentric simplices are the true generalization of triangles and to the study of other families of simplices like circumscriptible, isodynamic and isogonic simplices.
In order to go through this discussion, in the last section of the paper there are proposed 15 interesting problems which cover several possible ways of directing the research.

MSC:

51M04 Elementary problems in Euclidean geometries
51-02 Research exposition (monographs, survey articles) pertaining to geometry
51Mxx Real and complex geometry
51M20 Polyhedra and polytopes; regular figures, division of spaces
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