Napoleon points

In geometry, Napoleon points are a pair of special points associated with a plane triangle. It is generally believed that the existence of these points was discovered by Napoleon Bonaparte, the Emperor of the French from 1804 to 1815, but many have questioned this belief. The Napoleon points are triangle centers and they are listed as the points X(17) and X(18) in Clark Kimberling's Encyclopedia of Triangle Centers. The name "Napoleon points" has also been applied to a different pair of triangle centers, better known as the isodynamic points. Let ABC be any given plane triangle. On the sides BC, CA, AB of the triangle, construct outwardly drawn equilateral triangles DBC, ECA and FAB respectively. Let the centroids of these triangles be X, Y and Z respectively. Then the lines AX, BY and CZ are concurrent. The point of concurrence N1 is the first Napoleon point, or the outer Napoleon point, of the triangle ABC. The triangle XYZ is called the outer Napoleon triangle of the triangle ABC. Napoleon's theorem asserts that this triangle is an equilateral triangle. In Clark Kimberling's Encyclopedia of Triangle Centers, the first Napoleon point is denoted by X(17). The trilinear coordinates of N1: The barycentric coordinates of N1: Let ABC be any given plane triangle. On the sides BC, CA, AB of the triangle, construct inwardly drawn equilateral triangles DBC, ECA and FAB respectively. Let the centroids of these triangles be X, Y and Z respectively. Then the lines AX, BY and CZ are concurrent. The point of concurrence N2 is the second Napoleon point, or the inner Napoleon point, of the triangle ABC. The triangle XYZ is called the inner Napoleon triangle of the triangle ABC. Napoleon's theorem asserts that this triangle is an equilateral triangle. In Clark Kimberling's Encyclopedia of Triangle Centers, the second Napoleon point is denoted by X(18). The trilinear coordinates of N2: The barycentric coordinates of N2: Two points closely related to the Napoleon points are the Fermat-Torricelli points (ETC's X13 and X14).