
Journal for Geometry and Graphics 15 (2011), No. 2, 159168 Copyright Heldermann Verlag 2011 Some Theorems on Kissing Circles and Spheres Ken Morita Institute of Technology and Science, University of Tokushima, 21 Minamijyosanjimacho, Tokushima 7708506, Japan morita@frc.tokushimau.ac.jp When three circles, O_{1}, O_{2}, O_{3}, are tangent externally to each other, there are only two circles tangent to the original three circles. This is a special case of the Apollonius problem, and such circles are called the inner and outer Soddy circles. Given the outer Soddy circle S, we can construct the new Apollonian circle I_{1} that is tangent to S, O_{2}, and O_{3}. By the same method, we can construct new circles I_{2} tangent to S, O_{3}, and O_{1}, and I_{3} tangent to S, O_{1}, and O_{2}. These seven tangent circles are a subset of an Apollonian packing of circles. In this article, we describe a new inscribed circle tangent to the three pairs of common external tangents of diagonally placed circles, {O_{1}, I_{1}}, {O_{2}, I_{2}}, and {O_{3}, I_{3}}. Furthermore, we found that when two externally tangent triangles of the three circles {O_{1}, O_{2}, O_{3}} and {I_{1}, I_{2}, I_{3}} are constructed, the three diagonally joined lines of the two triangles are concurrent. These theorems are further generalized to the threedimensional case on nine tangent spheres. Focusing on visual representations, we established these theorems only by a synthetic method throughout this article. Keywords: Tangent circles and spheres, inversions of circles and spheres. MSC: 51M04; 51N10 [ Fulltextpdf (1193 KB)] for subscribers only. 