Napoleon's and van Aubel's Theorems

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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.

[more]

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.

[less]

Contributed by: Robert Dickau (March 2011)
Open content licensed under CC BY-NC-SA


Snapshots


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.



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send