Starting with the circle tangent to the three semicircles forming the arbelos, construct a chain of tangent circles , all tangent to one of the two small interior circles and to the large exterior one. This chain is called the Pappus chain (left figure).

In a Pappus chain, the distance from the center of the first inscribed circle to the bottom line is twice the circle's radius, from the second circle is four times the radius, and for the th circle is times the radius. Furthermore, the centers of the circles lie on an ellipse (right figure).

If , then the center and radius of the th circle in the Pappus chain are

			(1)
			(2)
			(3)

This general result simplifies to for (Gardner 1979). Further special cases when are considered by Gaba (1940).

The positions of the points of tangency for the first circle are

			(4)
			(5)
			(6)
			(7)
			(8)
			(9)

The diameter of the th circle is given by ()th the perpendicular distance to the base of the semicircle. This result was known to Pappus, who referred to it as an ancient theorem (Hood 1961, Cadwell 1966, Gardner 1979, Bankoff 1981). Note that this is also valid for the chain of tangent circles starting with and tangent to the two interior semicircles of the arbelos. The simplest proof is via inversive geometry.

Eliminating from the equations for and , the center of the circle , gives

(10)

Completing the square gives

(11)

which can be rearranged as

(12)

which is simply the equation of an ellipse having center and semimajor and semiminor axes and respectively. Since

(13)

and 1/2, so the ellipse has foci at the centers of the semicircles bounding the chain.

The circles tangent to the first arbelos semicircle and adjacent Pappus circles and have positions and sizes

			(14)
			(15)
			(16)

A special case of this problem with (giving equal circles forming the arbelos) was considered in a Japanese temple tablet (Sangaku problem) from 1788 in the Tokyo prefecture (Rothman 1998). In this case, the solution simplifies to

			(17)
			(18)
			(19)

Furthermore, the positions and radii of the three tangent circles surrounding this circle can also be found analytically, and are given by

			(20)
			(21)
			(22)
			(23)
			(24)
			(25)
			(26)
			(27)
			(28)

If divides in the golden ratio , then the circles in the chain satisfy a number of other special properties (Bankoff 1955).

In each arbelos, there are two Pappus chains and , with . For fixed , the line connecting the centers of and passes through the external center of similitude of the two smaller semicircles of the arbelos. The line connecting the point of tangency of and and the point of tangency of and passes through as well. Also the line connecting the point of tangency of and the large exterior semicircle (the smaller interior semicircle) and the point of tangency of and the large exterior semicircle (the smaller interior semicircle) passes through . This can be proven with circle inversion. In particular, since , the common tangent of and the large exterior semicircle passes through .

Portions of this entry contributed by Floor van Lamoen

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.

Cadwell, J. H. Topics in Recreational Mathematics. Cambridge, England: Cambridge University Press, 1966.

Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 103, 1888.

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. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 117, 1929.

Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85-91, May 1998.

Steiner, J. Jacob Steiner's gesammelte Werke, Band I. Bronx, NY: Chelsea, p. 47, 1971.

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van Lamoen, Floor and Weisstein, Eric W. "Pappus Chain." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PappusChain.html