Combining Computer Science and Math: Circumscribing a Circle about a Triangle Using the Geometry Applet

Abstract

Objective

This project has two objectives:

1. write a mathematical proof for the construction of a circle that circumscribes a triangle;
2. illustrate the proof with a dynamic figure created with the Geometry Applet.

Credits

Andrew Olson, Ph.D., Science Buddies

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Introduction

The illustration below shows a circle, with center at point D, circumscribed about triangle ABC. This is the circumcircle of the triangle, and point D is the circumcenter of the triangle. By definition, the points A, B, and C all fall on the circumference of the circle.

This project has two objectives:

1. write a mathematical proof for the construction of a circle circumscribed about a triangle;
2. illustrate the proof with a dynamic figure created with the Geometry Applet.

What is the Geometry Applet? It is a very cool program written by Professor David Joyce to illustrate an online version of Euclid's Elements. The applet creates dynamic diagrams in which you can manipulate the geometric figures by clicking and dragging on points. You program the applet much like creating a geometrical construction by hand, so as the points are dragged, all of the essential relationships in the diagram remain intact. It is an engaging and intuitive way to illustrate the generality of your proof. To see an example of the Geometry Applet in action, see any of these three projects:
Throwing You Some Curves: Is Red or Blue Longer?
Thinking in (Semi-)Circles: The Area of the Arbelos
Chain Reaction: Inversion and the Pappus Chain Theorem

To learn how to use the Geometry Applet to create your own dynamic diagrams, see:
Getting Started with the Geometry Applet

Terms and Concepts

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

• circumcircle,
• circumcenter,
• congruent triangles,
• bisection of a line segment.

• In general, there are three possible positions for the circumcenter of a triangle, relative to the triangle itself. What are the three positions? What property of the triangle is associated with each position of the circumcenter?

Bibliography

• The Math Forum at Drexel University has some good advice on how to build a mathematical proof:
  http://mathforum.org/library/drmath/view/54693.html
• There are many more examples in their FAQ section:
  http://mathforum.org/dr.math/faq/faq.proof.html
• For help with the programming the Geometry Applet, see:
  Getting Started with the Geometry Applet
• 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:
  http://aleph0.clarku.edu/~djoyce/java/elements/elements.html

Materials and Equipment

• For completing the proof manually, all you'll need is:
  • pencil,
  • paper,
  • compass, and
  • straightedge.
• For creating a dynamic diagram of the proof using the Geometry Applet, you will also need:
  • a computer,
  • a text editor (Notepad will work fine),
  • a Web browser program (e.g., Internet Explorer or Firefox).
• For a live demonstration along as part of your display, a laptop computer is recommended.

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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.

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Related Links

• Science Fair Project Guide
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• Computer Science Project Ideas
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