10182

# Napoleon's and van Aubel's Theorems

Starting with any triangle ABC, construct an equilateral triangle on each side, and then connect the centers of the triangles. The points form another equilateral triangle. This result is known as Napoleon's theorem.
For a quadrilateral ABCD, the similar process of constructing a square on each side and then joining the centers does not result in a square. However, the line segments joining the centers of opposite squares are always the same length and at right angles to each other. This is known as van Aubel's theorem.

### DETAILS

Snapshot 1: the triangle relationship holds even if the three points are collinear
Snapshot 3: the quadrilateral relationship holds even when the line segments do not intersect
M. Gardner, Chapters 5 and 14, Mathematical Circus, New York: Alfred A. Knopf, 1979.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.