| 释义 |
Soddy CirclesGiven three distinct points  ,  , and  , let three Circles be drawn, one centered about each point andeach one tangent to the other two. Call the Radii  ( ,  ,  ). Then theCircles satisfy
 | (1) |
 | (2) |
 | (3) |
Solving for the Radii then gives
The above Triangle has sides  ,  , and  , and Semiperimeter
 | (7) |
 | (8) |
 | (9) |
 | (10) |
 to opposite sides of the equation and noting that the above argument applies equally wellto  and  then gives
As can be seen from the first figure, there exist exactly two nonintersecting Circles which areTangent to all three Circles. These are called the inner and outer Soddy circles ( and  ,respectively), and their centers are called the inner and outer Soddy Points.
The inner Soddy circle is the solution to the Four Coins Problem. The center of the inner Soddy circle is theEqual Detour Point, and the center of the outer Soddy circle is the Isoperimetric Point (Kimberling 1994).
Frederick Soddy (1936) gave the Formula for finding the Radii of the Soddy circles ( ) given theRadii ( , 2, 3) of the other three. The relationship is
 | (14) |
 are the so-called Bends, defined as the signedCurvatures of the Circles. If the contacts are all external, the signs are all taken asPositive, whereas if one circle surrounds the other three, the sign of this circle is taken as Negative (Coxeter 1969). Using the Quadratic Formula to solve for  , expressing in terms of radii instead of curvatures, andsimplifying gives
 | (15) |
This Descartes.  However, Soddyalso extended it to Spheres. Gosper has further extended the result to  mutually tangent  -DHyperspheres, whose Curvatures satisfy
 | (16) |
 gives
 | (17) |
For (at least)  and 3, the Radical equals
 | (18) |
 is the Content of the Simplex whose vertices are the centers of the  independentHyperspheres. The Radicand can also become Negative, yielding anImaginary . For  , this corresponds to a sphere touching three large bowling balls and a small BB, all mutually tangent, which is an impossibility.
Bellew has derived a generalization applicable to a Circle surrounded by Circles which are, inturn, circumscribed by another Circle. The relationship is  | |  | (19) |
where
 | (20) |
References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 13-14, 1969.Elkies, N. D. and Fukuta, J. ``Problem E3236 and Solution.'' Amer. Math. Monthly 97, 529-531, 1990. Kimberling, C. ``Central Points and Central Lines in the Plane of a Triangle.'' Math. Mag. 67, p. 181, 1994. ``The Kiss Precise.'' Nature 139, 62, 1937. Soddy, F. ``The Kiss Precise.'' Nature 137, 1021, 1936. Vandeghen, A. ``Soddy's Circles and the De Longchamps Point of a Triangle.'' Amer. Math. Monthly 71, 176-179, 1964.
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