In geometry, Napoleon's theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward or all inward, the centres of those equilateral triangles themselves form an equilateral triangle.
The triangle thus formed is called the inner or outer Napoleon triangle. The difference in area of these two triangles equals the area of the original triangle.
The theorem is often attributed to Napoleon Bonaparte (1769–1821). Some have suggested that it may just date back to W. Rutherford's 1825 question published in The Ladies' Diary, four years after the French emperor's death., but the result is covered in three questions set in an examination for a Gold Medal at the University of Dublin in October, 1820, whereas Napoleon died the following May.
In the figure above, ABC is the original triangle. AZB, BXC, and CYA are equilateral triangles constructed on its sides' exteriors, and points L, M, and 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 the triangle LMN is equilateral is to observe that MN becomes CZ under a clockwise rotation of 30° around A and a homothety of ratio √3 with the same center, and that LN also becomes CZ after a counterclockwise rotation of 30° around B and a homothety of ratio √3 with the same center. The respective spiral similarities are A(√3,-30°) and B(√3,30°). That implies MN = LN and the angle between them must be 60°.
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 it is to be noted that 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 suffiently 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, 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.
Dublin Problems, October, 1820
In the Geometry paper, set on the second morning of the papers for candidates for the Gold Medal in the General Examination of the University of Dublin in October 1820, the following three problems appear.
- Question 10. Three equilateral triangles are thus constructed on the sides of a given triangle, A, B, D, the lines joining their centres, C, C', C" form an equilateral triangle. [The accompanying diagram shows the equilateral triangles placed outwardly.]
- Question 11. If the three equilateral triangles be constructed as in the last figure, the lines joining their centres will also form an equilateral triangle. [The accompanying diagram shows the equilateral triangles places inwardly.]
- Question 12. To investigate the relation between the area of the given triangle and the areas of these two equilateral triangles.
These problems are recorded in
- Dublin problems: a collection of questions proposed to the candidates for the gold medal at the general examinations, from 1816 to 1822 inclusive. Which is succeeded by an account of the fellowship examination, in 1823 (G. and W. B. Whittaker, London, 1823)
Question 1249 in the Gentleman's Diary; or Mathematical Repository for 1829 (so appearing in late 1828) takes up the theme, with solutions appearing in the issue for the following year. One of the solvers, T. S. Davies then generalized the result in Question 1265 that year, presenting his own solution the following year, drawing on a paper he had already contributed to the Philosophical Magazine in 1826. There are no cross-references in this material to that described above. However, there are several items of cognate interest in the problem pages of the popular almanacs both going back to at least the mid-1750s (Moss) and continuing on to the mid-1860s (Mason), as alluded to above.
As it happens, Napoleon's name is mentioned in connection with this result in no less a work of reference than Chambers's Encylopedia as early as 1867 (Vol. IX, towards the close of the entry on triangles).
- Another remarkable property of triangles, known as Napoleon's problem is as follows: if on any triangle three equilateral triangles be described, and the centres of gravity of these three be joined, the triangle thus formed is equilateral, and has its centre of gravity coincident with that of the original triangle.
But then the result had appeared, with proof, in a textbook by at least 1834 (James Thomson's Euclid).
Areas and sides of inner and outer Napoleon triangles
The area of the inner Napoleon triangle of a triangle with area is
where a, b, and 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 the inner triangle is "retrograde" and its algebraic area is the negative of this expression.
The area of the outer Napoleon triangle is
The relation between the latter two equations is that the area of an equilateral triangle equals the square of the side times
A family of Napoleon equilateral triangles
Given , construct three similar isosceles triangles either all outward, or all inward with base angle . Choose six points respectively on the rays such that:
Then và are both equilateral triangles
- Grünbaum 2012
- "Napoleon's Theorem - from Wolfram MathWorld". Mathworld.wolfram.com. 2013-08-29. Retrieved 2013-09-06.
- Weisstein, Eric W., "Spiral Similarity", MathWorld.
- For a visual demonstration see Napoleon's Theorem via Two Rotations at Cut-the-Knot.
- "Napoleon's Theorem" at MathPages.com.
- Alexander Bogomolny. "Proof #2 (an argument by symmetrization)". Cut-the-knot.org. Retrieved 2013-09-06.
- Cavallaro, V.G. (1949), "Per la storia dei teoremi attribuiti a Napoleone Buonaparte e a Frank Morley", Archimede 1: 286–287
- Scriba, Christoph J (1981). "Wie kommt 'Napoleons Satz' zu seinem namen?". Historia Mathematica 8 (4): 458–459. doi:10.1016/0315-0860(81)90054-9.
- Faifofer (1911), Elementi di Geometria (17th ed.), Venezia, p. 186, but the historical record cites various editions in different years. This reference is from (Wetzel 1992)
- http://solo.bodleian.ox.ac.uk/primo_library/libweb/action/dlDisplay.do?vid=OXVU1&docId=oxfaleph014134656 http://dbooks.bodleian.ox.ac.uk/books/PDFs/590315941.pdf> 22.8MB
- Dao Thanh Oai, A family of Napoleon triangles associated with the Kiepert configuration, The Mathematical Gazette, Published online: 13 March 2015
- Weisstein, Eric W. "Inner Napoleon Triangle." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/InnerNapoleonTriangle.html
- Coxeter, H.S.M., and Greitzer, Samuel L. 1967. Geometry Revisited, page 64.
- Weisstein, Eric W. "Outer Napoleon Triangle." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/OuterNapoleonTriangle.html
- Coxeter, H.S.M.; Greitzer, S.L. (1967). Geometry Revisited. New Mathematical Library 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, 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. Zbl 1264.01010.
|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.