Napoleon's Theorem
Yet Another Analytic Proof

Napoleon's theorem



J. A. Grzesik
Am Math Monthly, 123, October 2016

Let equilateral triangles be erected upon the sides of an arbitrary triangle, all on the exterior, or else all on the interior, denoted below by $\pm.\;$ Then it is a theorem, dubiously attributed to Napoleon Bonaparte and proved in a variety of ways, that the three lines connecting in sequence the centroids of these three equilateral triangles themselves form an equilateral triangle. The following short analytic proof may or may not have been overlooked.

Equip the triangle vertices with Cartesian coordinates $(x_i, y_i),\;$ $i=0,1,2,\;$ assigned consecutively from any one vertex as starting point and, for definiteness, in counterclockwise progression. The centroid of the exterior/interior equilateral triangle attached to the side running from $(x_i, y_i)\;$ to $(x_{i+1}, y_{i+1}),\;$ (the indices being taken modulo $3\;$ throughout) is readily seen to lie at

$\displaystyle\left(\begin{array}{cc} xc_i \\ yc_i\end{array}\right)=\left(\begin{array}{cc}\frac{x_i +x_{i+1}}{2}\pm\frac{y_i - y_{i+1}}{2\sqrt{3}}\\\frac{y_i + y_{i+1}}{2}\pm\frac{x_i -x_{i+1}}{2\sqrt{3}}\end{array}\right).$

A modest amount of manipulation suffices to segregate all terms making up the square of the distance linking $(xc_i, yc_i)\;$ to $(xc_{i+1}, yc_{i+1})\;$ into a category

$\displaystyle \frac{1}{3}\left[x^2_i +x^2_{i+1} +x^2_{i+2} -x_ix_{i+1} -x_{i+1}x_{i+2} -x_{i+2}x_i\right].$

quadratic in $x\;$ coordinates alone, a second, formally identical category having each $x^*\;$ replaced by its $y^*\;$ counterpart, and a third, mixed category

$\displaystyle \pm\frac{1}{\sqrt{3}}\left[x_i(y_{i+1} - y_{i+2})+x_{i+1}(y_{i+2} - y_i)+x_{i+2}(y_i - y_{i+1})\right]$

populated by terms bilinear in both $x\;$ and $y\;$ coordinates. One then verifies by inspection that each category is unchanged under index advance $i \rightarrow i +1,\;$ which shows that centroid locations $(xc_i, yc_i),\;$ $i=0,1,2\;$ occupy the vertices of an equilateral triangle, a distinct triangle accompanying each sign choice $\pm.\;$ Two alternate proofs, one purely synthetic and one analytic, are found on p. 38 of The Ladies’ Diary, Vol. 123, 1826.

Napoleon's Theorem

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