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Journal for Geometry and Graphics 15 (2011), No. 2, 159--168
Copyright Heldermann Verlag 2011

Some Theorems on Kissing Circles and Spheres

Ken Morita
Institute of Technology and Science, University of Tokushima, 2-1 Minami-jyosanjima-cho, Tokushima 770-8506, Japan

When three circles, O1, O2, O3, 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 I1 that is tangent to S, O2, and O3. By the same method, we can construct new circles I2 tangent to S, O3, and O1, and I3 tangent to S, O1, and O2. 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, {O1, I1}, {O2, I2}, and {O3, I3}. Furthermore, we found that when two externally tangent triangles of the three circles {O1, O2, O3} and {I1, I2, I3} are constructed, the three diagonally joined lines of the two triangles are concurrent. These theorems are further generalized to the three-dimensional 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

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