Napoleon's theorem: if equilateral triangles are constructed on the sides of any triangle (all outward or all inward), the centers of those equilateral triangles themselves form an equilateral triangle.
Proof: ( . . . . . . find better picture; label angles A, B, C, relabel letters a->c, b->a, c->b)
Beginning with an arbitrary triangle ABC, construct equilateral triangles a, b, c as shown. Construct a fourth equilateral triangle, d, as shown, with sides equal to those of triangle b. Triangle d is oriented the same as triangle b. (Proof: trace a path along the four line segments consisting of sides of triangles d, a, a, and b. Treating left turns as negative angles and right turns as positive angles, the turns, in sequence, are B-180, 120, and 60-B, which add up to zero.)
If both the triangle ABC and the equilateral triangle b are rotated counterclockwise about c through an angle of 120 degrees, side AC will line up with a different side of equilateral triangle c, and so the image of triangle b is concident with triangle d, proving line segments dc and cb are equal in length. Similarly, rotating triangle ABC and equilateral triangle b clockwise through an angle of 120 degrees shows line segments ab and ad are equal in length. By constructing the dark blue line ac, we see it bisects angles a and c, which are each 120 degrees, so angles cab, abc, and bca are all 60 degrees, proving the theorem.
Mathpages: Napoleon's Theorem is an extremely detailed and complete treatment of Napoleon's Theorem, including a delightful tessilation of the plane that results from the construction of a simple proof of the theorem.
Cut-the-knot: Napoleon's Theorem is the introductory page to a dozen other pages with various observations and generalizations of the theorem.
Mathworld: Napoleon's Theorem illustrates the theorem and very well describes one particular generalization: instead of equilateral triangles constructed on the sides of the original triangle, construct an arbitrary triangle on one side, then similar triangles with coincident sides in cyclic order. When the centroids of the constructed triangles are connected, the result is a fourth triangle similar to the three that were constructed on the sides of the original triangle.
Planet Math: Napoleon's Theorem gives a proof using the complex number plane.
Wikipedia: Napoleon's theorem
Jim Loy's pages: Napoleon's Theorem
Summary of geometrical theorems
Van Aubel's Theorem: Line segments connecting centers of squares on opposite sides of a quadrilateral are perpendicular and equal in length.
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