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Project Summary

Difficulty	8
Time required	Short (several days)
Prerequisites	You should either currently be taking or have already completed a first course in geometry. You must understand the concept and method of a mathematical proof.
Material Availability	Readily available
Cost	Very Low (under $20)
Safety	No issues

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Sponsored by a generous grant from Motorola

Abstract

Objective

Objective: Prove that AC = EF.

Introduction

Figure 1 below shows 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, AB, 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.

Figure 2: The arbelos.

Notes on How to Manipulate the Diagram

The diagram is illustrated using the Geometry Applet (by kind permission of the author, see Bibliography). If you have any questions about the applet, send us an email at: scibuddy@sciencebuddies.org. With the help of the applet, you can manipulate the diagram by dragging points.

In order to take advantage of this applet, be sure that you have enabled Java on your browser. If you disable Java, or if your browser is not Java-capable, then the diagram will still appear, but as a plain, still image.

If you click on a point in the diagram, you can usually move it in some way. The free points, usually colored red, can be freely dragged about, and as they move, the rest of the diagram (except the other free points) will adjust appropriately. Sliding points, usually colored orange, can be dragged about like the free points, except their motion is limited to either a straight line, a circle, a plane, or a sphere, depending on the point. Other points can be dragged to translate the entire diagram. If a pivot point appears, usually colored green, then the diagram will be rotated and scaled around that pivot point. (Note that diagrams will often use only one or two of the above types of points.)

You can't drag a point off the diagram, but frequently parts of the diagram will be moved off as you drag other points around. If you type r or the space key while the cursor is over the diagram, then the diagram will be reset to its original configuration.

You can also lift the diagram off the page into a separate window. When you type u or return the diagram is moved to its own window. Typing d or return while the cursor is over the original window will return the diagram to the page. Note that you can resize the floating window to make the diagram larger.

An interesting property of the arbelos is the "twin circles" discovered by Archimedes (see Figure 2, another dynamic diagram). If you draw circles tangent to the line CD and inscribed within the arbelos, the circles will be congruent. Archimides proved this in his Book of Lemmas (proposition 5).

Figure 2: The "twin circles" of Archimedes.

This project centers around a curious property of the arbelos that involves the twin circles. The line segment EF in Figure 3 passes through the center of the right-hand twin circle. Its endpoint F is the point of tangency where the right-hand twin circle touches the smaller semicircle of the arbelos. The endpoint E is the intersection of the extension of CD and the line from F through the center of the right-hand twin circle. Curiously enough, line segment EF is congruent with line segment AC, the diameter of the other semicircle inside the largest semicircle of the arbelos. Can you prove that EF = AC?

Figure 3: Prove that AC = EF.

Terms, Concepts and Questions to Start Background Research

To do this project, you should do research that enables you to understand the following terms and concepts:

• right triangles,
• similar triangles,
• the arbelos,
• mathematical proof.

Bibliography

• This problem, along with some very interesting background material on the arbelos, is presented in the following reference:
  Boas, H.P., 2006. "Reflections on the Arbelos," American Mathematical Monthly 113:236–249 [accessed March 3, 2006] http://www.math.tamu.edu/~harold.boas/preprints/arbelos.pdf.
• For more information on the arbelos and Archimedes twin circles, see:
  • Weisstein, E.W., 1999. "Arbelos." From MathWorld—A Wolfram Web Resource. [accessed March 3, 2006] http://mathworld.wolfram.com/Arbelos.html.
  • Bogomolny, A., 2006. "The Book of Lemmas: Proposition 5," Cut-the-Knot, Geometry section [accessed March 3, 2006] http://www.cut-the-knot.org/Curriculum/Geometry/BookOfLemmas/BOL5.shtml.
• The Math Forum at Drexel University has some good advice on how to build a mathematical proof:
  
  • Peterson, 1999. "How To Build a Proof," Ask Dr. Math, The Math Forum @Drexel University [accessed March 3, 2006] http://mathforum.org/library/drmath/view/54693.html
  • Staff, 1999–2006. "About Proofs," Ask Dr. Math, The Math Forum @Drexel University [accessed March 3, 2006] http://mathforum.org/dr.math/faq/faq.proof.html
• The Geometry Applet was kindly provided to Science Buddies by its author, Professor David Joyce, who wrote it for illustrating an amazing online edition of Euclid's Elements. Book IV, Proposition 5, and Book VI, Proposition 8 will be helpful:
  • Joyce, D., 1998. "Book IV, Proposition 5," Euclid's Elements, Department of Mathematics and Computer Science, Clark University [accessed January 21, 2006] http://aleph0.clarku.edu/~djoyce/java/elements/bookIV/propIV5.html
  • Joyce, D., 1998. "Book VI, Proposition 8," Euclid's Elements, Department of Mathematics and Computer Science, Clark University [accessed January 21, 2006] http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI8.html

Materials and Equipment

To do this experiment you will need the following materials and equipment:

• pencil,
• paper,
• compass, and
• straightedge.

• computer with Internet access, Web broswer and a text editor (like Notepad).

Experimental Procedure

1. Do your background research,
2. organize your known facts, and
3. spend some time thinking about the problem and you should be able to come up with the proof.

Variations

• For a more basic project, see
  Thinking in (Semi-)Circles: The Area of the Arbelos.
• For another project that uses geometric inversion in a circle, see
  Chain Reaction: Inversion and the Pappus Chain Theorem.
• If you'd like to try making your own dynamic diagram with the Geometry Applet, check out these Science Buddies links:
  • Geometry Applet How-to Pages
  • Circles, Tangent Lines and Triangles

• For more science project ideas in this area of science, see Pure Mathematics Project Ideas.

Credits

• Boas, H.P., 2006. "Reflections on the Arbelos," American Mathematical Monthly 113:236–249 [accessed March 3, 2006] http://www.math.tamu.edu/~harold.boas/preprints/arbelos.pdf.
  
  
• Andrew Olson, Ph.D., Science Buddies
  
• Professor David Joyce, for the Geometry Applet

Last edit date: 2006-04-19 23:59:59

Career Focus

If you like this project, you might enjoy exploring careers in Pure Mathematics.

	Mathematician Mathematicians are part of an ancient tradition of searching for patterns, conjecturing, and figuring out truths based on rigorous deduction. Some mathematicians focus on purely theoretical problems, with no obvious or immediate applications, except to advance our understanding of mathematics, while others focus on applied mathematics, where they try to solve problems in economics, business, science, physics, or engineering.			Statistician Statisticians use the power of math and probability theory to answer questions that affect the lives of millions of people. They tell educators which teaching method works best, tell policy-makers what levels of pesticides are acceptable in fresh fruit, tell doctors which treatment works best, tell builders which type of paint is the most durable. They are employed in virtually every type of industry imaginable, from engineering, manufacturing, and medicine to animal science, food production, transportation, and education. Everybody needs a statistician!
	Math Teacher Math teachers love mathematics and understand it well, but much more than that, they enjoy sharing their enthusiasm for the language of numbers with students. They use a variety of tools and techniques to help students grasp abstract concepts and show them that math describes the world around them. By helping students conquer fears and anxieties about math, teachers can open up many science and technology career possibilities for students. Teachers make a difference that lasts a lifetime!			Economist Every country has resources—people, land, raw materials, capital, and machinery—and economists study how those resources are distributed to create the goods that people buy, and the services people need or want. In their studies, economists monitor economic trends and collect data on things like energy costs, inflation, interest rates, exchange rates, business cycles, taxes, and employment levels. Based on their analysis of this data, they develop forecasts of economic activity so that businesses and governments can better plan for the future.

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