Napoleon points
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.
[1] 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.
In Clark Kimberling's Encyclopedia of Triangle Centers, the first Napoleon point is denoted by X(17).
In Clark Kimberling's Encyclopedia of Triangle Centers, the second Napoleon point is denoted by X(18).
If instead of constructing lines joining the equilateral triangles' centroids to the respective vertices one now constructs lines joining the equilateral triangles' apices to the respective vertices of the triangle, the three lines so constructed are again concurrent.
The results regarding the existence of the Napoleon points can be generalized in different ways.
These centers can be thought as the vertices of isosceles triangles erected on the sides of triangle ABC with the base angles equal to π/6 (30 degrees).
The generalizations seek to determine other triangles that, when erected over the sides of △ABC, have concurrent lines joining their external vertices and the vertices of △ABC.
This generalization asserts the following:[4] If the common base angle is θ, then the vertices of the three triangles have the following trilinear coordinates.
Moreover, the locus of N as the base angle θ varies between −π/2 and π/2 is the conic
This conic is a rectangular hyperbola and it is called the Kiepert hyperbola in honor of Ludwig Kiepert (1846–1934), the mathematician who discovered this result.
[4] This hyperbola is the unique conic which passes through the five points A, B, C, G, O.
[5] The concurrence of the lines AX, BY, CZ holds even in much relaxed conditions.
The following result states one of the most general conditions for the lines AX, BY, CZ to be concurrent.
They observe that Napoleon Bonaparte was a bit of a mathematician with a great interest in geometry.
However, they doubt whether Napoleon knew enough geometry to discover the theorem attributed to him.
The Ladies' Diary was an annual periodical which was in circulation in London from 1704 to 1841.
The result appeared as part of a question posed by W. Rutherford, Woodburn.
However, there is no reference to the existence of the so-called Napoleon points in this question.
Christoph J. Scriba, a German historian of mathematics, has studied the problem of attributing the Napoleon points to Napoleon in a paper in Historia Mathematica.
