Pappus Chain
In the Arbelos, construct a chain of Tangent Circles starting with the CircleTangent to the two small interior semicircles and the large exterior one. Then the distance from the center of the firstInscribed Circle to the bottom line is twice the Circle's Radius, from the secondCircle is four times the Radius, and for the 
th Circle is 
times the Radius. The centers ofthe Circles lie on an Ellipse, and the Diameter of the th Circle 
is (
)th Perpendicular distance to the base of the Semicircle. This result was known toPappus, 
who referred to it as an ancient theorem (Hood 1961, Cadwell 1966, Gardner 1979, Bankoff 1981). Thesimplest proof is via Inversive Geometry.
If 
, then the radius of the th circle in the pappus chain is
This equation can be derived by iteratively solving the Quadratic Formula generated by Descartes Circle Theoremfor the radius of the Soddy Circle. This general result simplifies to
for 
(Gardner 1979). Further special cases when 
are considered by Gaba (1940).If 
divides 
in the Golden Ratio 
, then the circles in the chain satisfy a number of other special properties(Bankoff 1955).
References
Bankoff, L. ``The Golden Arbelos.'' Scripta Math. 21, 70-76, 1955. Bankoff, L. ``Are the Twin Circles of Archimedes Really Twins?'' Math. Mag. 47, 214-218, 1974. Bankoff, L. ``How Did Pappus Do It?'' In The Mathematical Gardner (Ed. D. Klarner). Boston, MA: Prindle, Weber, and Schmidt, pp. 112-118, 1981. Gaba, M. G. ``On a Generalization of the Arbelos.'' Amer. Math. Monthly 47, 19-24, 1940. Gardner, M. ``Mathematical Games: The Diverse Pleasures of Circles that Are Tangent to One Another.'' Sci. Amer. 240, 18-28, Jan. 1979. Hood, R. T. ``A Chain of Circles.'' Math. Teacher 54, 134-137, 1961. Johnson, R. A. Advanced Euclidean Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 117, 1929.
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