Soddy Circles and David Eppstein's Centers: What Are They?
A Mathematical Droodle
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Copyright © 19962018 Alexander Bogomolny
Soddy Circles and David Eppstein's Centers
The applet suggests that for three circles externally tangent to each other, there exist two circles that touch all circles in the triplet. One of the circles touches the triplet internally, the other touches all of them externally. The circles bear the name of Frederick Soddy (18771956), a distinguished chemist and economist, a 1921 Nobel prize winner, who made the circles famous by publishing the result along with a poem "The kiss precise" in Nature, 137 (1936), p. 1021. The circles are correspondingly called the outer and the inner Soddy circles. Their centers are also known as the Soddy points of the triangle formed by the centers of the triplet.
(This is a particular case of the more general Apollonius' problem that asks for a circle tangent to the given three circles (or lines, or passing through given points) in an arbitrary configuration relative to each other.)
The formula that links the radii of four circles that touch each other externally was known to Descartes [Pedoe, p. 158]. It is given in terms of the curvatures, bends in Soddy's terminology. For a circle, the curvature is the reciprocal of its radius, and Descartes' formula appears as ([Pedoe, p. 157, Coxeter, 1.57])
2(ε_{1}^{2} + ε_{2}^{2} + ε_{3}^{2} + ε_{4}^{2}) = (ε_{1} + ε_{2} + ε_{3} + ε_{4})^{2},  
where ε_{i} = 1/R_{i} is the curvature of the ith circle (R_{i} being its radius),
If one of the curvatures is taken with the sign "", then the formula corresponds to the case when one of the circles touches the rest internally, i.e. when one of them  the outer Soddy circle  encloses the other three.
In 1826, the famous geometer Jacob Steiner rediscovered the formula. In 1842, it was proven independently by Philip Beecroft, an English amateur mathematician. Frederick Soddy's contribution appeared in 1936.
Thus, the Soddy circles have a long history, in the course of which they were under close attention of several exceptional people. This is then so much more remarkable that one interesting property of the configuration has been discovered as late as 2001. In 2001, D. Eppstein, of the Geometry Junkyard fame, published the following observation (see also a partial online version):
(1)  Four touching circles, when taken two by two, define two points of tangency and, therefore, a straight line. There are three such lines. The three lines are concurrent. 
This is true for both inner and outer configuration, so there are two points of note, known now as Eppstein's points.
Eppstein derives his result as a particular case of a 3D configuration of four spheres. Any four mutually tangent spheres determine six points of tangency. The six points are naturally divided into three pairs of opposite tangencies, i.e. tangencies, in which one is defined by two spheres distinct from the pair of the spheres defining the other.
Lemma
[AltshillerCourt, p. 231].
The three lines through the opposite points of tangency of any four mutually tangent spheres in R^{3} are concurrent. 
Proof
Let the four given spheres S_{i},
t_{i}_{j} = (ε_{i}O_{i} + ε_{j}O_{j})/(ε_{i} + ε_{j}). 
This is a weighted average of the centers of the two spheres. A similar average for the centers of all four spheres
(2)  M = (ε_{1}O_{1} + ε_{2}O_{2} + ε_{3}O_{3} + ε_{4}O_{4})/(ε_{1} + ε_{2} + ε_{3} + ε_{4}) 
could also be represented as
M = ((ε_{1}O_{1} + ε_{2}O_{2})t_{12} + (ε_{3}O_{3} + ε_{4}O_{4})t_{34})/(ε_{1} + ε_{2} + ε_{3} + ε_{4}), 
which means that M lies on the line joining the opposite tangencies t_{12} and t_{34}. By the same argument, it lies on the other two lines.
(1) follows from Lemma when the centers of the four spheres happen to be coplanar.
If one starts with a triangle ABC, then there exists a unique triplet C_{A}, C_{B}, C_{C} of pairwise tangent circles centered at the vertices of the triangle. Let a, b, c be the side lengths of ΔABC (
R_{A} = s  a, R_{B} = s  b, R_{C} = s  c, 
where s is the semiperimeter of ΔABC, s = (a + b + c)/2. The points of tangency t_{AB}, t_{AC}, and t_{BC} are exactly the points where the incircle of ΔABC touches its sides. The three lines At_{BC}, Bt_{AC}, and Ct_{AB} are concurrent at the Gergonne point Ge.
(3)  Ge = (ε_{A}A + ε_{B}B + ε_{C}C)/(ε_{A} + ε_{B} + ε_{C}) 
because, for example,
(ε_{A}A + ε_{B}B + ε_{C}C)/(ε_{A} + ε_{B} + ε_{C}) = (ε_{A}A + (ε_{B} + ε_{C})t_{BC})/(ε_{A} + ε_{B} + ε_{C}) 
so that the point in (2) lies on the line joining A and t_{BC}, and similarly for the other two lines. (2) suggests that a relationship between M and Ge should not be unexpected. Indeed, among other things, Eppstein proves the following
Theorem
Let S be the center of the inner Soddy circle. Then S, M, and Ge are collinear. 
Proof
The proof follows by combining (2) and (3):
(2') 

It's known that S, Ge, and I, the incenter of ΔABC are collinear. M therefore lies on the same line. By symmetry, the same is true of the outer Eppstein point.
Reference
 N. AltshillerCourt, Modern Pure Solid Geometry, Chelsea, 2nd ed, 1964
 H. S. M. Coxeter, Introduction to Geometry, John Wiley & Sons, 1961
 H. S. M. Coxeter, The Problem of Apollonius, Am Math Monthly, 75 (1968), pp. 515
 D. Eppstein, Tangent Spheres and Triangle Centers, Amer Math Monthly 108 (2001), pp. 6366
 A. Oldknow, The EulerGergonneSoddy triangle of a triangle, Amer Math Monthly 103 (1996), pp. 319329
 D. Pedoe, Geometry: A Comprehensive Course, Dover, 1970
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny