Abstract (eng)
A triangle possesses many distinguished points. The most well-known of these points are the incenter, the excenters, the circumcenter, the centroid, and the orthocenter. This thesis deals, among other things, with the generalization of these points to three- and higher-dimensional space. Furthermore, the generalization of Euler's line and Feuerbach's circle to three- and higher-dimensional space is addressed. In two- and three-dimensional space, the proofs were mostly demonstrated geometrically as well as algebraically. The incenter, circumcenter, and centroid can be naturally generalized to higher-dimensional space. However, the excenters and the orthocenter are more complex. In three- and higher-dimensional space, the altitudes of a simplex generally do not intersect at a single point. However, there exists another point, the Monge point, which exhibits many properties of the orthocenter in higher-dimensional space. For example, in higher-dimensional space, the Monge point lies on the line through the centroid and the circumcenter of a simplex, which also allows for the generalization of Euler's line. Equally interesting is the possible existence of excenters in higher-dimensional space. For example, a tetrahedron possesses between four and seven exspheres, each touching the supporting plane of the lateral faces. In general, simplices have different types of circumspheres, the existence of which depends on the size of the lateral faces. The set of existing circumspheres is limited but exhibits gaps in odd dimensions. No information about whether such gaps occur in even dimensions could be found.