Napoleon's theorem claims that the centers A', B', C', of the equilateral triangles A''BC, AB''C, ABC'', erected on the sides (either all inwardly or all outwardly) of a given triangle ABC form an equilateral triangle. Napoleon's theorem admits several generalizations (see Douglas' Theorem, Kiepert's theorem). One generalization has been discovered in the 1955 by Adriano Barlotti and is now known as the Napoleon-Barlotti Theorem. The theorem was rediscovered by L. Gerber in 1980.
The clue to the generalization is suggested by V. Thébault's theorem of 1937: On the sides of parallelogram ABCD erect squares -- all either on the outside or the inside of the parallelogram. Their centers then form another square.
The Napoleon-Barlotti Theorem extends both Napoleon's and Thébault's theorem to an arbitrary N-gon: On the sides of an affine-regular N-gon construct regular N-gons (all either the same orientations as the base N-gon or all opposite to it). Then the centers of these regular N-gon form a regular N-gon.
A polygon is affine-regular if it's an affine image of a regular polygon. Any triangle is an affine image of an equilateral triangle. Parallelograms (and only they) are affine images of a square. In general, 2N-sided affine-regular polygon is necessarily a paragon, but the class of affine-regular polygons is both more general (there are affine-regular polygons with an odd number of sides) and more specific (not all 2N-sided paragons are affine regular.) For the latter, the reason is in that an affine 2D transformation (modulo a translation) is defined by 4 parameters a11, a12, a21, a22 - usually shown in a matrix form. The are many more degrees of freedom in forming 2N-paragons. (Relatively recently the affine-regular polygons have been studied by Duane DeTemple and Matthew Hudelson.)
The applet below illustrates the Napoleon-Barlotti Theorem. The four scrollbars control the values of the four parameters a11, a12, a21, a22.
The dashed polygon is the regular one to which the affine transform applies to produce the affine-regular polygon shown with thick boundary.
- A. Barlotti, Una proprietà degli n-agoni che si ottengono trasformando in una affinità un n-agono regolare, Boll. Un. Mat. Ital. (3) 10 (1955) 96-98
- D. DeTemple, M. Hudelson, Square-Banded Polygons and Affine Regularity, The American Mathematical Monthly, Vol. 108, No. 2 (Feb., 2001), pp. 100-114
- L. Gerber, Napoleon's Theorem and the Parallelogram Inequality for Affine-Regular Polygons, The American Mathematical Monthly, Vol. 87, No. 8 (Oct., 1980), pp. 644-648
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