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
Objective: Prove that AC = EF.
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
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Last edit date: 2013-04-25
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 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. [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).
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Experimental Procedure
- Do your background research,
- organize your known facts, and
- spend some time thinking about the problem and you should be able to come up with the proof.
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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:
Share your story with Science Buddies!
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