  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.
