Some Properties of Napoleon's ConfigurationNapoleon'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. The applet below serves to illustrate a couple of properties of Napoleon's configuration.
First PropertyThe centroids of Napoleon's triangle A'B'C' and of the reference triangle ABC coincide. The simplest proof I am aware of makes use of complex numbers, see a separate discussion. The proof holds both for the outer and the inner Napoleon's triangles. (Both cases can be observed in the applet which originally displays the construction of the outer triangle. This is the one where the equilateral triangles are formed outside ΔABC. The inner case can be observed by change the orientation of ΔABC. This is done by dragging one of the vertices across the opposite side line.) It follows that the centroids of the two Napoleon triangles - the outer and the inner - coincide. This fact was mentioned in the classical [Geometry Revisited, p. 65], [Complex Numbers and Geometry, p. 113] or [MathematicalGems, p. 40]. The two books, however, failed to mention that the same point also serves as the centroid of the base triangle ABC. Second PropertyLines AA'', BB'', CC'' meet in a point - one (either inner or outer) of Fermat's points of ΔABC.The three lines form equal angles of 60° or 120°, depending on which ones are looked at. This is equivalent to the claim that the circumcircles of triangles ABC'', AB''C, and A''BC are concurrent. This implies the next property. Third PropertyThe sides of the Napoleon triangle serve as perpendicular bisectors of the lines joining Fermat's point to the vertices of ΔABC. This follows from the fact that, by the construction, the vertices of Napoleon's triangle are the centers of the circumcircles of triangles ABC'', AB''C, and A''BC while the segments AF, BF, CF (F being the Fermat point) are the common chords of the circles taken two at a time. Fourth PropertyThe circumcricles of Napoleon's triangles are concurrent; in an acute triangle they meet at Fermat's point. As such, they admit a triangle with one vertex on each of the circles whose sides pass through the vertices of the base triangle. This is actually a porism: the starting point for the construction of that triangle can be chosen arbitrarily. The triangle is always equilateral. References
 Napoleon's Theorem
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