# Taking Off on a Tangent

## Abstract

This is an interesting geometry project that goes back to the time of Archimedes, the famous Greek mathematician. You can combine this mathematical project with computer science and take this ancient problem into the twenty-first century with a dynamic diagram using the geometry applet.## Objective

Prove that*AC*=

*EF*.

## Credits

Andrew Olson, Ph.D., Science Buddies

Professor David Joyce, for the Geometry Applet

- Boas, H.P., 2006. "Reflections on the Arbelos,"
*American Mathematical Monthly*113:236–249. Retrieved March 3, 2006 from http://www.math.tamu.edu/~harold.boas/preprints/arbelos.pdf.

## Share your story with Science Buddies!

I Did This Project! Please log in and let us know how things went.Last edit date: 2013-05-31

## Introduction

**Important: You will need the current version of Java installed on your computer for this project.**If you do not have Java, some figures and images in this project may not display properly in your browser. To check for the current version of Java on your machine, visit Java.

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 1.**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 an image may display stating "This plugin is disabled" or the diagram may 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 and Concepts

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. Retrieved March 3, 2006 from http://mathworld.wolfram.com/Arbelos.html - Bogomolny, A., 2006. "The Book of Lemmas: Proposition 5," Cut-the-Knot, Geometry section. Retrieved March 3, 2006 from 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. Retrieved March 3, 2006 from http://mathforum.org/library/drmath/view/54693.html
- Staff, 1999–2006. "About Proofs," Ask Dr. Math, The Math Forum @Drexel University. Retrieved March 3, 2006 from 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. Retrieved January 21, 2006 from 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. Retrieved January 21, 2006 from 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
- Straightedge

If you want to make your own dynamic figure to illustrate your proof using the Geometry Applet, you'll also need:

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

## Share your story with Science Buddies!

I Did This Project! Please log in and let us know how things went.## Experimental Procedure

- Do your background research.
- Organize your known facts.
- Spend some time thinking about the problem and you should be able to come up with the proof.

## Share your story with Science Buddies!

I Did This Project! Please log in and let us know how things went.## Variations

For a more basic project, see:

For another project that uses geometric inversion in a circle, see:

If you'd like to try making your own dynamic diagram with the Geometry Applet, check out these Science Buddies links:

## Share your story with Science Buddies!

I Did This Project! Please log in and let us know how things went.## Ask an Expert

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