Some theorems on kissing circles and spheres
type: Item
instance of: scholarly article
zbMATH Open document ID: 1255.51010
author: Ken Morita
publication date: 2012-09-25T00:00:00Z
full work available at URL: jgg15012.htm
Mathematics Subject Classification ID: 51M04
Mathematics Subject Classification ID: 51N10
Mathematics Subject Classification ID: 51N15
Mathematics Subject Classification ID: 52C26
zbMATH DE Number: 6086379
zbMATH Keywords: tangent circles and spheres
zbMATH Keywords: Apollonian circles
zbMATH Keywords: inversions of circles and spheres
MaRDI profile type: Publication
title: Some theorems on kissing circles and spheres
published in: Journal for Geometry and Graphics
review text: Let \(O_1, O_2\) and \(O_3\) be circles externally tangent to each other, and suppose that the circle \(S\) contains \(O_1,O_2\) and \(O_3\) in its interior and is tangent to all of \(O_1, O_2\) and \(O_3\). Let \(I_1, I_2\) and \(I_3\) be the three additional circles tangent to \(S, O_2\) and \(O_3\), to \(S, O_1\) and \(O_3\) and to \(S, O_1\) and \(O_2\), respectively. Denote by \(s_i\) and \(t_i\) the two common external tangents of \(O_i\) and \(I_i\), \(i=1,2,3\). Then there exists a circle tangent to all of \(s_1, t_1, s_2, t_2, s_3\) and \(t_3\). Moreover, let \(\triangle A_1 A_2 A_3\) denote the triangle whose edges are given by the outer common external tangents of \(O_1,O_2\) and \(O_3\) and, similarly, let \(\triangle B_1 B_2 B_3\) be given by the outer common external tangents of \(I_1,I_2\) and \(I_3\). Finally, let \(l_1,l_2\) and \(l_3\) be the straight lines joining opposite vertices of \(\triangle A_1 A_2 A_3\) and \(\triangle B_1 B_2 B_3\). Then \(l_1,l_2\) and \(l_3\) pass through a common point.NEWLINENEWLINEThe elementary proof of the above is based on inversions of circles and spheres. An analogous claim is shown for three-dimensional tangent spheres.
reviewed by: Christian Richter
description: scientific article; zbMATH DE number 6086379