Napoleon points

Template:Short description 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.

Definition of the points

First Napoleon point

First Napoleon point

Let Template:Math be any given plane triangle. On the sides Template:Mvar of the triangle, construct outwardly drawn equilateral triangles Template:Math respectively. Let the centroids of these triangles be Template:Mvar respectively. Then the lines Template:Mvar are concurrent. The point of concurrence Template:Math is the first Napoleon point, or the outer Napoleon point, of the triangle Template:Math.

The triangle Template:Math is called the outer Napoleon triangle of Template:Math. 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 Template:Math:$$\begin{array}{cc}& \csc (A+\frac{\pi }{6}):\csc (B+\frac{\pi }{6}):\csc (C+\frac{\pi }{6})\\ =& \sec (A-\frac{\pi }{3}):\sec (B-\frac{\pi }{3}):\sec (C-\frac{\pi }{3})\end{array}$$
• The barycentric coordinates of Template:Math:$$a\csc (A+\frac{\pi }{6}):b\csc (B+\frac{\pi }{6}):c\csc (C+\frac{\pi }{6})$$

Second Napoleon point

Second Napoleon point

Let Template:Math be any given plane triangle. On the sides Template:Mvar of the triangle, construct inwardly drawn equilateral triangles Template:Math respectively. Let the centroids of these triangles be Template:Mvar respectively. Then the lines Template:Mvar are concurrent. The point of concurrence Template:Math is the second Napoleon point, or the inner Napoleon point, of Template:Math.

The triangle Template:Math is called the inner Napoleon triangle of Template:Math. 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 Template:Math:$$\begin{array}{cc}& \csc (A-\frac{\pi }{6}):\csc (B-\frac{\pi }{6}):\csc (C-\frac{\pi }{6})\\ =& \sec (A+\frac{\pi }{3}):\sec (B+\frac{\pi }{3}):\sec (C+\frac{\pi }{3})\end{array}$$
• The barycentric coordinates of Template:Math:$$a\csc (A-\frac{\pi }{6}):b\csc (B-\frac{\pi }{6}):c\csc (C-\frac{\pi }{6})$$

Two points closely related to the Napoleon points are the Fermat-Torricelli points (ETC's X(13) and X(14)). 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 points of concurrence are called the Fermat-Torricelli points, sometimes denoted Template:Math and Template:Math. The intersection of the Fermat line (i.e., that line joining the two Fermat-Torricelli points) and the Napoleon line (i.e., that line joining the two Napoleon points) is the triangle's symmedian point (ETC's X(6)).

Generalizations

The results regarding the existence of the Napoleon points can be generalized in different ways. In defining the Napoleon points we begin with equilateral triangles drawn on the sides of Template:Math and then consider the centers Template:Mvar of these triangles. These centers can be thought as the vertices of isosceles triangles erected on the sides of triangle ABC with the base angles equal to Template:Pi/6 (30 degrees). The generalizations seek to determine other triangles that, when erected over the sides of Template:Math, have concurrent lines joining their external vertices and the vertices of Template:Math.

Isosceles triangles

A point on the Kiepert hyperbola.

Kiepert hyperbola of Template:Math. The hyperbola passes through the vertices Template:Mvar, the orthocenter Template:Mvar and the centroid Template:Mvar of the triangle.

This generalization asserts the following:

If the three triangles Template:Math, constructed on the sides of the given triangle Template:Math as bases, are similar, isosceles and similarly situated, then the lines Template:Mvar concur at a point Template:Mvar.

If the common base angle is Template:Mvar, then the vertices of the three triangles have the following trilinear coordinates. $$\begin{array}{cccccc}X=& -\sin \theta & :& \sin (C+\theta )& :& \sin (B+\theta )\\ Y=& \sin (C+\theta )& :& -\sin \theta & :& \sin (A+\theta )\\ Z=& \sin (B+\theta )& :& \sin (A+\theta )& :& -\sin \theta \end{array}$$

The trilinear coordinates of Template:Mvar are $$\csc (A+\theta ):\csc (B+\theta ):\csc (C+\theta )$$

A few special cases are interesting.

Value of Template:Mvar	The point Template:Mvar
Template:Tmath	Template:Mvar, the centroid of Template:Math
Template:Tmath	Template:Mvar, the orthocenter of Template:Math
Template:Tmath	The Vecten points
Template:Tmath	Template:Math, the first Napoleon point X(17)
Template:Tmath	Template:Math, the second Napoleon point X(18)
Template:Tmath	Template:Math, the first Fermat–Torricelli point X(13)
Template:Tmath	Template:Math, the second Fermat–Torricelli point X(14)
$\{\begin{array}{cc}-A& \text{if }A<\pi /2\\ \pi -A& \text{if }A>\pi /2\end{array}$	The vertex Template:Mvar
$\{\begin{array}{cc}-B& \text{if }B<\pi /2\\ \pi -B& \text{if }B>\pi /2\end{array}$	The vertex Template:Mvar
$\{\begin{array}{cc}-C& \text{if }C<\pi /2\\ \pi -C& \text{if }C>\pi /2\end{array}$	The vertex Template:Mvar

Moreover, the locus of Template:Mvar as the base angle Template:Mvar varies between −Template:Pi/2 and Template:Pi/2 is the conic

$$\frac{\sin (B-C)}{x}+\frac{\sin (C-A)}{y}+\frac{\sin (A-B)}{z}=0.$$

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. This hyperbola is the unique conic which passes through the five points Template:Mvar.

Similar triangles

Generalization of Napoleon point: A special case

The three triangles Template:Math erected over the sides of the triangle Template:Math need not be isosceles for the three lines Template:Mvar to be concurrent.

If similar triangles Template:Math are constructed outwardly on the sides of any triangle Template:Math then the lines Template:Mvar are concurrent.

Arbitrary triangles

The concurrence of the lines Template:Mvar holds even in much relaxed conditions. The following result states one of the most general conditions for the lines Template:Mvar to be concurrent.

If triangles Template:Math are constructed outwardly on the sides of any triangle Template:Math such that

$$\mathrm{\angle }CBX=\mathrm{\angle }ABZ,\mathrm{\angle }ACY=\mathrm{\angle }BCX,\mathrm{\angle }BAZ=\mathrm{\angle }CAY;$$

then the lines Template:Mvar are concurrent.

The point of concurrency is known as the Jacobi point.

A generalization of Napoleon point

History

Coxeter and Greitzer state the Napoleon Theorem thus: If equilateral triangles are erected externally on the sides of any triangle, their centers form an equilateral triangle. 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 earliest recorded appearance of the result embodied in Napoleon's theorem is in an article in The Ladies' Diary appeared in 1825. 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.

VII. Quest.(1439); by Mr. W. Rutherford, Woodburn." Describe equilateral triangles (the vertices being either all outward or all inward) upon the three sides of any triangle ABC: then the lines which join the centers of gravity of those three equilateral triangles will constitute an equilateral triangle. Required a demonstration."

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.

See also

• Triangle center
• Triangle conic
• Napoleon's theorem
• Napoleon's problem
• Van Aubel's theorem
• Fermat point

Further reading

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