New extensions of Napoleon's theorem to higher dimensions

Mowaffaq Hajja, Horst Martini and Margarita Spirova

Abstract: If equilateral triangles are erected outwardly on the sides of any given triangle, then the circumcenters of the three erected triangles form an equilateral triangle. This statement, known as Napoleon's theorem, and the configuration involved, usually called the Torricelli configuration of the initial triangle, were generalized to $d$-dimensional simplices ($d\ge 3$) in [MW]. It is obvious that for $d\ge 3$ regular $d$-simplices cannot be erected on the facets of an arbitrary initial $d$-simplex $S$. Thus, instead of erecting such simplices, the authors of [MW] used a related sphere configuration which also occurs in the planar situation. In the present paper, we give new $d$-dimensional analogues, mainly based on a higher dimensional Torricelli configuration constructed with the help of segments on lines through isogonal points and vertices of $S$. Interesting further properties of $d$-dimensional Torricelli configurations are obtained, too. [MW] Martini, H.; Weissbach, B.: Napoleon's theorem with weights in $n$-space. Geom. Dedicata {\bf 74} (1999), 213--223.

Keywords: centroid, circumcenter, equiareal simplex, Fermat-Torricelli point, incenter, isogonal point, Napoleon simplex, Napoleon's theorem, Torricelli configuration

Classification (MSC2000): 51M04; 51M20; 51N20; 52B11

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