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Napolean's theorem

Napolean Bonaparte, Emperor of France (March 1804 - April 1814, March 1815 - June 1815) and amateur mathematician is credited with discovering the theorem known as Napolean's theorem. Click and drag any of the blue points to change the shape and orientation of the triangle ABC. What do you observe?

Complete the following: "If equilateral triangles are constructed on the sides of any triangle, either all outward, or all inward, the centroids of those equilateral triangles ..."