Napoleon's theorem
If equilateral triangles are constructed on the sides of any triangle, either all outward or all inward, the lines connecting the centroids of those equilateral triangles themselves form an equilateral triangle.
The triangle thus formed is called the inner or outer Napoleon triangle. The difference in the areas of the outer and inner Napoleon triangles equals the area of the original triangle.
The theorem is attributed to Napoleon Bonaparte (1769–1821) though this has been questioned.
This short video presents a visual proof - Youtube: Napoleon's theorem - proof
Further material:
Wolfram.com - Napoleon's theorem
Mathpages.com - Napoleon's theorem
Cut the Knot - Napoleon's theorem, two simple proofs
Napoleon's theorem made simple
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