Chain Reaction: Inversion and the Pappus Chain Theorem *
Abstract
Here is a challenging problem for anyone with an interest in geometry. This project requires background research to solve it, but it is an excellent illustration of visual thinking in mathematics.Figure 1 below shows a series of circles (iC₁, iC₂, iC₃, ..., iC₃₀), inscribed inside an arbelos. What is an arbelos? The arbelos is the white region in the figure, bounded by three semicircles. The diameters of the three semicircles are all on the same line segment, AC, and each semicircle is tangent to the other two. The arbelos has been studied by mathematicians since ancient times, and was named, apparently, for its resemblance to the shape of a round knife (called an arbelos) used by leatherworkers in ancient times. The chain of inscribed circles is sometimes called a Pappus Chain, for Pappus of Alexandria, who studied and wrote about it in the 4th century A.D. The inscribed circles are tangent to one another, and to the boundaries of the arbelos. That is, iC₁ is tangent to each of the three semicircles that form the boundary of the arbelos, while each successive circle is tangent to the preceding one and to two of the semicircles that bound the arbelos (note that, in its default position, the figure illustrates just one of three possible configurations of the chain). Pappus proved a theorem (which he called "ancient"), which states that the height, h_{n}, of the center of the n^{th} inscribed circle, iC_{n}, above the line segment AC is equal to n times the diameter of iC_{n}.
Figure 1. Chain of circles inscribed in the arbelos.
Pappus' proof, relying solely on Euclidean geometry, ran over many pages. The modern proof is much simpler and uses the powerful method of circle inversion, invented in the 1820's by Jacob Steiner. Your goal is to gain a sufficient understanding of the principles of circle inversion and their application to the arbelos so that you can prove Pappus' Theorem:
The references in the Bibliography will help you get started learning more about circle inversion.
Objective
Prove that the height, h_{n}, of the center of the n^{th} inscribed circle, iC_{n}, above the line segment AC is equal to n times the diameter of iC_{n}.
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Andrew Olson, Science Buddies, author,
Prof. Harold P. Boas and Alex Bogomolny, for their inspiring and insightful publications on the subject
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Last edit date: 20170728
Bibliography
Circle inversion. This is the heart of the matter. Here are a couple of links to get you started on your research.
Here is an excellent introduction to the concept and method of circle inversion:
Another reference is: Eric W. Weisstein. Inversion. From MathWorldA Wolfram Web Resource.
The arbelos. There are several websites with more information on the fascinating properties of the arbelos.
 Eric W. Weisstein. Arbelos. From MathWorldA Wolfram Web Resource. http://mathworld.wolfram.com/Arbelos.html
 Boas, H. P. Reflections on the Arbelos. Preprint based on the Nineteenth Annual RoseHulman Undergraduate Mathematics Conference, March 15, 2002. Oct. 26, 2004. http://www.math.tamu.edu/~harold.boas/preprints/arbelos.pdf
Alex Bogomolny has a wideranging mathematics website. Here is a link to his page on the arbelos:
Pappus of Alexandria. You'll find a brief biography of Pappus here:
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Variations
 The Introduction notes that "in its default position, the figure illustrates just one of three possible configurations of the chain." A second configuration can be obtained by sliding point B closer to point C, which has the effect of "flipping" the arbelos. It is as if the chain of circles in the original figure were descending toward point A. The theorem also holds for a chain of inscribed circles descending toward point B. Construct a figure which illustrates this configuration.
 The arbelos has many other fascinating features. Perhaps your research will turn up some other feature which you wish to explore for your science fair project.
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