Napoleon's theorem

In the figure above, △ABC is the original triangle. △AZB, △BXC, △CYA are equilateral triangles constructed on its sides' exteriors, and points L, M, N are the centroids of those triangles. The theorem for outer triangles states that triangle △LMN (green) is equilateral.

A quick way to see that △LMN is equilateral is to observe that MN becomes CZ under a clockwise rotation of 30° around A and a homothety of ratio ${\displaystyle {\sqrt {3}}}$ with the same center, and that LN also becomes CZ after a counterclockwise rotation of 30° around B and a homothety of ratio ${\displaystyle {\sqrt {3}}}$ with the same center. The respective spiral similarities[3] are ${\displaystyle A({\sqrt {3}},-30^{\circ }),\ B({\sqrt {3}},30^{\circ }).}$ That implies MN = LN and the angle between them must be 60°.[4]

There are in fact many proofs of the theorem's statement, including a synthetic (coordinate-free) one,[5] a trigonometric one,[6] a symmetry-based approach,[7] and proofs using complex numbers.[6]

Extract from the 1826 Ladies' Diary giving geometric and analytic proofs

The theorem has frequently been attributed to Napoleon, but several papers have been written concerning this issue[8][9] which cast doubt upon this assertion (see (Grünbaum 2012)).

The following entry appeared on page 47 in the Ladies' Diary of 1825 (so in late 1824, a year or so after the compilation of Dublin examination papers). This is an early appearance of Napoleon's theorem in print, and Napoleon's name is not mentioned.

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 centres of gravity of those three equilateral triangles will constitute an equilateral triangle. Required a demonstration."

Since William Rutherford was a very capable mathematician, his motive for requesting a proof of a theorem that he could certainly have proved himself is unknown. Maybe he posed the question as a challenge to his peers, or perhaps he hoped that the responses would yield a more elegant solution. However, it is clear from reading successive issues of the Ladies' Diary in the 1820s, that the Editor aimed to include a varied set of questions each year, with some suited for the exercise of beginners.

Plainly there is no reference to Napoleon in either the question or the published responses, which appeared a year later in 1826, though the Editor evidently omitted some submissions. Also, Rutherford himself does not appear amongst the named solvers after the printed solutions, although from the tally a few pages earlier it is evident that he did send in a solution, as did several of his pupils and associates at Woodburn School, including the first of the published solutions. Indeed, the Woodburn Problem Solving Group, as it might be known today, was sufficiently well known by then to be written up in A Historical, Geographical, and Descriptive View of the County of Northumberland ... (2nd ed. Vo. II, pp. 123–124). It had been thought that the first known reference to this result as Napoleon's theorem appears in Faifofer's 17th Edition of Elementi di Geometria published in 1911,[10] although Faifofer does actually mention Napoleon in somewhat earlier editions. But this is moot because we find Napoleon mentioned by name in this context in an encyclopaedia by 1867. What is of greater historical interest as regards Faifofer is the problem he had been using in earlier editions: a classic problem on circumscribing the greatest equilateral triangle about a given triangle that Thomas Moss had posed in the Ladies Diary in 1754, in the solution to which by William Bevil the following year we might easily recognize the germ of Napoleon's Theorem - the two results then run together, back and forth for at least the next hundred years in the problem pages of the popular almanacs: when Honsberger proposed in Mathematical Gems in 1973 what he thought was a novelty of his own, he was actually recapitulating part of this vast, if informal, literature.

It might be as well to recall that a popular variant of the Pythagorean proposition, where squares are placed on the edges of triangles, was to place equilateral triangles on the edges of triangles: could you do with equilateral triangles what you could do with squares - for example, in the case of right triangles, dissect the one on the hypotenuse into those on the legs? Just as authors returned repeatedly to consider other properties of Euclid's Windmill or Bride's Chair, so the equivalent figure with equilateral triangles replacing squares invited - and received - attention. Perhaps the most majestic effort in this regard is William Mason's Prize Question in the Lady's and Gentleman's Diary for 1864, the solutions and commentary for which the following year run to some fifteen pages. By then, this particular venerable venue - starting in 1704 for the Ladies' Diary and in 1741 for the Gentleman's Diary - was on its last legs, but problems of this sort continued in the Educational Times right into the early 1900s.

The centers of both the inner and outer Napoleon triangles coincide with the centroid of the original triangle. This coincidence was noted in Chambers's Encyclopaedia in 1867, as quoted above. The entry there is unsigned. P. G. Tait, then Professor of Natural Philosophy in the University of Edinburgh, is listed amongst the contributors, but J. U. Hillhouse, Mathematical Tutor also at the University of Edinburgh, appears amongst other literary gentlemen connected for longer or shorter times with the regular staff of the Encyclopaedia. However, in Section 189(e) of An Elementary Treatise on Quaternions,[14] also in 1867, Tait treats the problem (in effect, echoing Davies' remarks in the Gentleman's Diary in 1831 with regard to Question 1265, but now in the setting of quaternions):

If perpendiculars be erected outwards at the middle points of the sides of a triangle, each being proportional to the corresponding side, the mean point of their extremities coincides with that of the original triangle. Find the ratio of each perpendicular to half the corresponding side of the old triangle that the new triangle may be equilateral.

Tait concludes that the mean points of equilateral triangles erected outwardly on the sides of any triangle form an equilateral triangle. The discussion is retained in subsequent editions in 1873 and 1890, as well as in his further Introduction to Quaternions [15] jointly with Philip Kelland in 1873.

The area of the inner Napoleon triangle of a triangle with area △ is

$${\displaystyle {\text{Area(inner)}}=-{\frac {\triangle }{2}}+{\frac {\sqrt {3}}{24}}(a^{2}+b^{2}+c^{2})\geq 0,}$$

where a, b, c are the side lengths of the original triangle, with equality only in the case in which the original triangle is equilateral, by Weitzenböck's inequality. However, from an algebraic standpoint[16] the inner triangle is "retrograde" and its algebraic area is the negative of this expression.[17]

The area of the outer Napoleon triangle is[18]

$${\displaystyle {\text{Area(outer)}}={\frac {\triangle }{2}}+{\frac {\sqrt {3}}{24}}(a^{2}+b^{2}+c^{2}).}$$

Analytically, it can be shown[6] that each of the three sides of the outer Napoleon triangle has a length of

$${\displaystyle {\text{Side(outer)}}={\sqrt {{a^{2}+b^{2}+c^{2} \over 6}+{{\sqrt {(a+b+c)(a+b-c)(a-b+c)(-a+b+c)}} \over {2{\sqrt {3}}}}}}.}$$

The relation between the latter two equations is that the area of an equilateral triangle equals the square of the side times ${\displaystyle {\sqrt {3}}/4.}$

• Napoleon points
• Lemoine's problem
• Napoleon's problem
• Van Aubel's theorem

• Coxeter, H.S.M.; Greitzer, S.L. (1967). Geometry Revisited. New Mathematical Library. Vol. 19. Washington, D.C.: Mathematical Association of America. pp. 60–65. ISBN 978-0-88385-619-2. Zbl 0166.16402.
• Grünbaum, Branko (2012), "Is Napoleon's Theorem Really Napoleon's Theorem?", American Mathematical Monthly, 119 (6): 495–501, doi:10.4169/amer.math.monthly.119.06.495, S2CID 42928783, Zbl 1264.01010
• Wetzel, John E. (April 1992). "Converses of Napoleon's Theorem" (PDF). The American Mathematical Monthly. 99 (4): 339–351. doi:10.2307/2324901. JSTOR 2324901. Zbl 1264.01010. Archived from the original (PDF) on 2014-04-29.

Wikimedia Commons has media related to Napoleon's theorem.

• Napoleon's Theorem and Generalizations, at Cut-the-Knot
• To see the construction, at instrumenpoche
• Napoleon's Theorem by Jay Warendorff, The Wolfram Demonstrations Project.
• Weisstein, Eric W. "Napoleon's Theorem". MathWorld.
• Napoleon's Theorem and some generalizations, variations & converses at Dynamic Geometry Sketches
• Napoleon's Theorem, Two Simple Proofs
• Infinite Regular Hexagon Sequences on a Triangle (generalization of Napoleon's Theorem) by Alvy Ray Smith.

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