Napoleon's theorem

Napoleon's theorem states that the centers of equilateral triangles erected consistently on the sides of any triangle form an equilateral triangle, for both inward and outward constructions.

Equilateral triangle: A triangle with three equal sides and three equal angles of 60 degrees. The constructed triangles are equilateral, and their centers become the theorem's vertices.
Triangle: A polygon with three sides and three interior angles. The theorem begins with an arbitrary triangle as its base figure.
External Napoleon theorem: The variant in which equilateral triangles are erected outward on all three sides of a triangle. It gives one of the two orientation choices covered by the theorem.
Napoleon Bonaparte: French military and political leader who became emperor from 1804 to 1814 and again briefly in 1815. The theorem bears his name, though a connection between him and its discovery is unverified.
Centroid: The point where a triangle's three medians intersect; it is also the triangle's center of mass when density is uniform. The theorem commonly uses the centroids of the erected equilateral triangles.
Circumcenter: The point equidistant from a triangle's three vertices, where its perpendicular bisectors meet. Some formulations use the circumcenters of the erected equilateral triangles instead of their centroids.
Internal Napoleon theorem: The variant in which equilateral triangles are erected inward on all three sides of a triangle. It gives the inward counterpart to the outward construction.
William Rutherford: A Scottish mathematician known for calculating many digits of pi and contributing to mathematical education. A nineteenth-century publication by Rutherford is among the early printed sources associated with the theorem.
Complex plane: A coordinate plane in which each point is represented by a complex number. Complex coordinates encode the side rotations and make the center-distance equality tractable.
Incenter: The point equidistant from a triangle's sides, where its angle bisectors meet. The incenter is another possible center, but it does not generally yield Napoleon's equilateral result.

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