On Miquel's theorem and inversions in normed planes
type: Item
instance of: scholarly article
title: On Miquel's theorem and inversions in normed planes
zbMATH Open document ID: 1210.51005
author: Margarita Spirova
published in: Monatshefte für Mathematik
publication date: 2010-10-27T00:00:00Z
review text: A normed plane \(\mathbb M\) is a 2-dimensional real vector space equipped with a norm. If the unit circle \(\mathcal C\) of all vectors of norm 1 in \(\mathbb M\) is strictly convex and smooth, then \textit{G. Ewald} [Abh. Math. Semin. Univ. Hamb. 30, 179--187 (1967; Zbl 0147.19802)] showed geometrically that the collection of all homothethic images of \(\mathcal C\) plus all Euclidean lines extended by a new point \(\infty\) form the circles of a spherical Möbius plane. The case where \(\mathcal C\) is an ellipse is characterized by the validity of Miquel's theorem, a well-known result which can be traced back to \textit{B. Hesselbach} [Abh. Math. Semin. Hamb. Univ. 9, 265--271 (1933; Zbl 0007.07303)]. In view of the correspondence between strictly convex, smooth normed planes and Möbius planes it is natural to consider Miquel's configuration in normed planes. \textit{H. Martini} and the author [Discrete Math. 309, 5158--5168 (2009; Zbl 1186.52007)] showed that Miquel's configuration closes in a strictly convex normed plane if the considered circles have equal radii. In the paper under review, the author extends a result of this kind to all normed planes without the assumption of strict convexity. Since circles can now intersect in segments, the formulation of an analogue to Miquel's configuration is rather involved. To this end the author introduces the notion of conjugate points with respect to two circles of equal radii. Four points \(p_i\), \(i=1,2,3,4\), in a cyclic order on a circle \(x+\lambda\mathcal{C}\) determine four further circles of radius \(\lambda\), whose centres \(y_i\) are obtained as conjugate points of \(x\) with respect to the consecutive circles \(p_i+\lambda\mathcal{C}\) and \(p_{i+1}+\lambda\mathcal{C}\). It is then shown that the four points that are the conjugates to the \(p_i\) with respect to consecutive pairs \(y_i+\lambda\mathcal{C}\) and \(y_{i+1}+\lambda\mathcal{C}\) of these four circles are on a circle with the same radius \(\lam…
reviewed by: Günter F. Steinke
Mathematics Subject Classification ID: 51B10
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 52A21
zbMATH DE Number: 5806775
zbMATH Keywords: finite-dimensional Banach space
zbMATH Keywords: normed plane
zbMATH Keywords: Miquel's theorem
zbMATH Keywords: Möbius plane
zbMATH Keywords: inversion
MaRDI profile type: Publication
full work available at URL: s00605-009-0153-8
OpenAlex ID: W1991807111
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DOI: 10.1007/S00605-009-0153-8
description: scientific article; zbMATH DE number 5806775