Monge Points, Euler Lines, and Feuerbach Spheres in Minkowski Spaces

Abstract

It is surprising, but an established fact that the field of Elementary Geometry referring to normed spaces (= Minkowski spaces) is not a systematically developed discipline. There are many natural notions and problems of elementary and classical geometry that were never investigated in this more general framework, although their Euclidean subcases are well known and this extended viewpoint is promising. An example is the geometry of simplices in non-Euclidean normed spaces; not many papers in this direction exist. Inspired by this lack of natural results on Minkowskian simplices, we present a collection of new results as non-Euclidean generalizations of well-known fundamental properties of Euclidean simplices. These results refer to Minkowskian analogues of notions like Euler line, orthocentricity, Monge point, and Feuerbach sphere of a simplex in a normed space. In addition, we derive some related results on polygons (instead of triangles) in normed planes.

An Analogue of Thébault's Theorem Linking the Radical Center of Four Spheres with the Insphere and the Monge Point of a Tetrahedron

“…All hyperplanes, each of which passes through the centroid of a (n − 2)face of F and perpendicular to the opposite edge of this face, have a common point. This point is called the Monge point of the simplex F [4].…”

“…There are six planes, each of which passes through the midpoint of the edge of the tetrahedron T perpendicular to its opposite edge. These six planes have a common point, which is called the Monge point of T [4,7,8].…”

Geometry of Simplices in Minkowski Spaces

“…In [16] the Euler line and the Feuerbach-or 2(d + 1)-sphere associated to a circumcenter M of a d-simplex T have been studied. The Euler line is the straight line containing M , the centroid G, the associated Feuerbach center F M (the center of the aforementioned sphere), the associated Monge point N M , and and the so-called associated complementary point P M .…”

“…The Euler line is the straight line containing M , the centroid G, the associated Feuerbach center F M (the center of the aforementioned sphere), the associated Monge point N M , and and the so-called associated complementary point P M . We recall from [16] that the point N M is the point of concurrence of certain lines, namely those which are well-defined by the centroid of a (d − 2)-face (or ridge) of T and the midpoint of the opposite edge. Furthermore, the point P M is the point of concurrence of certain other lines, each containing a vertex of T and parallel to a line defined by the centroid of the opposite facet and M , provided the latter two points do not coincide.…”