Universität zu Köln

Struve, Rolf ORCID: 0000-0001-6915-5623 and Struve, Horst (2016). An axiomatic analysis of the Droz-Farny Line Theorem. Aequ. Math., 90 (6). S. 1201 - 1219. BASEL: SPRINGER BASEL AG. ISSN 1420-8903

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Abstract

We analyze an elementary theorem of Euclidean geometry, the Droz-Farny Line Theorem, from the point of view of the foundations of geometry. We start with an elementary synthetic proof which is based on simple properties of the group of motions. The proof reveals that the Droz-Farny Line Theorem is a special case of the Theorem of Goormatigh which is, in turn, a special case of the Counterpairing Theorem of Hessenberg. An axiomatic analysis in the sense of Hilbert [14] and Bachmann [2] leads to a study of different versions of the theorems (e.g., of a dual version or of an absolute version, which is valid in absolute geometry) and to a new axiom system for the associated very general plane absolute geometry (the geometry of pencils and lines). In the last section the role of the theorems in the foundations of geometry is discussed.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Struve, Rolf UNSPECIFIED orcid.org/0000-0001-6915-5623 UNSPECIFIED Struve, Horst UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-254244
DOI:	10.1007/s00010-016-0430-2
Journal or Publication Title:	Aequ. Math.
Volume:	90
Number:	6
Page Range:	S. 1201 - 1219
Date:	2016
Publisher:	SPRINGER BASEL AG
Place of Publication:	BASEL
ISSN:	1420-8903
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language GEOMETRY; PLANES Multiple languages Mathematics, Applied; Mathematics Multiple languages
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/25424