by: Doris Santarone

**Essay #2: Napolean's Triangle**

__Background__

Napoleon's theorem states that if we construct equilateral triangles on the sides of any triangle (all outward or all inward), the centers of those equilateral triangles themselves form an equilateral triangle. Below is an illustration of Napolean's Triangle, the first with the equilateral triangles constructed outward and the second with the equilateral triangles constructed inward.

Click Here for this GSP Sketch

Click Here for this GSP Sketch

This theorem was first seen in the 1826 article by Dr. W. Rutherford in "The Ladies Diary." The theorem is attributed to Napolean Bonaparte, but there is no known evidence supporting the therorem's connection to him.

__Proofs__

I will show a few proofs of Napolean's Theorem:

**Extension**

The figure formed by Napolean's Triangle also tesselates the plane. Click here to explore the tesselation.