Monge points, Euler lines, and Feuerbach spheres in Minkowski spaces

TL;DR

This paper explores classical geometric concepts like Monge points, Euler lines, and Feuerbach spheres within Minkowski (normed) spaces, extending Euclidean properties to a more general non-Euclidean setting.

Contribution

It introduces the first systematic study of these Euclidean geometric notions in Minkowski spaces, providing new generalizations and results for simplices and polygons.

Findings

Generalized Euler line, Monge point, and Feuerbach sphere in Minkowski spaces.

New properties of simplices in non-Euclidean normed spaces.

Results on polygons in normed planes.

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.