On the Miquel point of simplices
type: Item
instance of: scholarly article
title: On the Miquel point of simplices
zbMATH Open document ID: 1147.51014
DOI: 10.4171/EM/98
author: Lothar Heinrich
published in: Elemente der Mathematik
publication date: 2008-09-08T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_pdf.php
review text: Let \(S:=S({\mathbf x}_0,{\mathbf x}_1,\dots,{\mathbf x}_d)\) be a \(d\)-simplex with positive volume of the real Euclidean \(d\)-space, \(d\geq 2\); if one point is marked on each of the \(d(d+1)/2\) edges of \(S\) and a sphere \(M_i\) is drawn through each vertex \({\mathbf x}_i\) and the points marked on the \(d\) edges which meet in \({\mathbf x}_i\), then these \(d+1\) Miquel spheres \(M_i\), \(i=0,1,\dots,d\), all meet in a point \(M\) (Miquel's point). For this multidimensional generalization of Miquel's theorem (\(d=2\)) the author gives a rigorous analytical proof. As a by-product the author obtains a family of upper bounds of Gram's determinant. Two figures accompany the elegant and careful considerations.
reviewed by: Rolf Riesinger
Mathematics Subject Classification ID: 51N20
zbMATH DE Number: 5322349
zbMATH Keywords: Miquel sphere
zbMATH Keywords: Miquel point
zbMATH Keywords: Gram's determinant
zbMATH Keywords: Hadamard's inequality
zbMATH Keywords: Brocard point
MaRDI profile type: Publication
cites work: Q5583621
cites work: Pivot theorems in n-space
OpenAlex ID: W2016265138
description: scientific article; zbMATH DE number 5322349