Steiner Chain
Given two nonconcentric Circles with one interior to the other, if small TangentCircles can be inscribed around the region between the two Circles such that the finalCircle is Tangent to the first, the Circles form a Steiner chain.
The simplest way to construct a Steiner chain is to perform an Inversion on a symmetrical arrangement on 
circlespacked between a central circle of radius 
and an outer concentric circle of radius 
(Wells 1991). In this arrangement,
 | (1) |
 | (2) |
To transform the symmetrical arrangement into a Steiner chain, take an Inversion Center which is a distance 
from thecenter of the symmetrical figure. Then the radii 
and 
of the outer and center circles become
 |  |  | (3) |
 | |  | (4) |
respectively. Equivalently, a Steiner chain results whenever the Inversive Distance between the two original circles isgiven by
 | |  | (5) |
|  | (6) |
(Coxeter and Greitzer 1967).
Steiner's Porism states that if a Steiner chain is formed from one starting circle, then a Steiner chain is alsoformed from any other starting circle.
See also Arbelos, Coxeter's Loxodromic Sequence of Tangent Circles, Hexlet, Pappus Chain, Steiner's Porism
References
Coxeter, H. S. M. ``Interlocking Rings of Spheres.'' Scripta Math. 18, 113-121, 1952. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 87, 1969. Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 124-126, 1967. Forder, H. G. Geometry, 2nd ed. London: Hutchinson's University Library, p. 23, 1960. Gardner, M. ``Mathematical Games: The Diverse Pleasures of Circles that Are Tangent to One Another.'' Sci. Amer. 240, 18-28, Jan. 1979. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 113-115, 1929. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 51-54, 1990.
Weisstein, E. W. ``Plane Geometry.'' Mathematica notebook PlaneGeometry.m.
- Fermat 4n+1 Theorem
- Fermat Compositeness Test
- Fermat Conic
- Fermat Difference Equation
- Fermat Diophantine Equation
- Fermat Equation
- Fermat-Euler Theorem
- Fermatian
- Fermat-Lucas Number
- Fermat Number
- Fermat Number (Lucas)
- Fermat Point
- Fermat Polynomial
- Fermat Prime
- Fermat Pseudoprime
- Fermat Quotient
- Fermat's Algorithm
- Fermat's Congruence
- Fermat's Conjecture
- Fermat's Factorization Method
- Fermat's Last Theorem
- Fermat's Lesser Theorem
- Fermat's Little Theorem
- Fermat's Little Theorem Converse
- Fermat Spiral Inverse Curve