Napoleon's theorem

• 1. Introduction

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  Napoleon's theorem sits at an odd crossroads between pure geometry and contested history. Draw any triangle you like - scalene, obtuse, lopsided, it does not matter. Build an equilateral triangle on each of its three sides, pointing outward. Then mark the center of each of those three new triangles. Connect those three center points, and something unexpected happens: they form a perfect equilateral triangle. Every time. No exceptions.

  The same holds in reverse. Build the equilateral triangles pointing inward, and again their centers form an equilateral triangle. The outer version and the inner version differ in size, but by a precise and elegant amount: the area gap between them equals exactly the area of the original triangle you started with.

  Who discovered this? That question has occupied mathematicians for nearly two centuries. The theorem carries the name of the French emperor Napoleon Bonaparte, yet the evidence linking him to it is thin. A gold medal examination at the University of Dublin in October 1820 included the result as three consecutive problems - and Napoleon died the following May. The question of what he actually had to do with this theorem leads into a story about almanacs, schoolmasters, an unsolved puzzle of mathematical attribution, and a result that kept resurfacing independently across nearly a century of popular mathematical literature.

• 2. The Geometry Itself

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  The centroid of each outward equilateral triangle is the point that anchors the whole construction. Label the centroids L, M, and N. The theorem for outer triangles says that triangle LMN is equilateral, regardless of what the original triangle looks like.

  One elegant way to see why this must be true uses spiral similarities. One equilateral triangle becomes another under a clockwise rotation of 30 degrees around one vertex combined with a scaling factor, and the same transformation applies from a different vertex with a counterclockwise rotation. Those two spiral similarities imply that the relevant distances are equal and that the angle between them is 60 degrees, which is exactly what an equilateral triangle requires.

  That is one proof, but the theorem admits many more. Mathematicians have established the result through synthetic coordinate-free reasoning, through trigonometry, through symmetry arguments, and through the algebra of complex numbers. The multiplicity of proof paths reflects how deeply the result connects to the geometry of triangles, not a narrow technical accident.

  A further structural fact emerges when both the inner and outer Napoleon triangles are considered together. The centers of both triangles coincide with the centroid of the original triangle. That shared center was noted in Chambers's Encyclopaedia in 1867. P. G. Tait, then Professor of Natural Philosophy at the University of Edinburgh, treated the problem in his Elementary Treatise on Quaternions in that same year, framing it as a question about perpendiculars erected at the midpoints of sides.

• 3. A Name Without a Clear Owner

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  Howard Eves suggested that the theorem and a related construction problem were actually discovered by Lorenzo Mascheroni, born in 1750 and died in 1800, who was a friend and adviser to Napoleon. According to Eves, Mascheroni let the Emperor claim the results for himself. If true, that would push the origin back before the Dublin examination of October 1820.

  The problem appeared in the Ladies' Diary of 1825 - meaning it was compiled in late 1824 - posed by William Rutherford of Woodburn. His phrasing describes exactly the theorem: equilateral triangles on the three sides of any triangle, vertices all outward or all inward, and the lines joining the centers of gravity forming an equilateral triangle. Napoleon's name appears nowhere in Rutherford's question or in the responses published the following year in 1826.

  William Rutherford was, by the account of the historical record, a very capable mathematician. Why he asked readers to prove something he could certainly have proved himself is unknown. He may have intended it as a challenge, or he may have hoped a more elegant solution would emerge from his peers. The Woodburn Problem Solving Group he led was sufficiently well regarded to be written up in a regional geographical survey of Northumberland.

  The first known printed reference to the result as Napoleon's theorem appears to have been in Chambers's Encyclopaedia in 1867, in the entry on triangles. That entry is unsigned, though the Encyclopaedia lists P. G. Tait and J. U. Hillhouse, Mathematical Tutor at the University of Edinburgh, among its contributors at various times. Faifofer's Elementi di Geometria had been thought to contain the first such named reference, in its 17th edition of 1911, but Faifofer mentions Napoleon in earlier editions, and the Chambers's entry predates both.

• 4. The Almanac Tradition

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  The Ladies' Diary began publishing in 1704, and the Gentleman's Diary in 1741. For roughly a century and a half, these almanacs served as the primary venue where working mathematicians and talented amateurs exchanged problems and solutions.

  Thomas Moss posed a classic problem about circumscribing the greatest equilateral triangle about a given triangle in the Ladies' Diary in 1754. William Bevil's solution the following year contained, as later readers recognized, the germ of what would become Napoleon's theorem. The two results then circulated together, back and forth, through the problem pages of popular almanacs for at least the next hundred years.

  In the Gentleman's Diary for 1829 - published in late 1828 - Question 1249 returned to the theme. Solutions appeared the following year, and one solver, T. S. Davies, then generalized the result in Question 1265, presenting his own solution the year after that. Davies drew on a paper he had contributed to the Philosophical Magazine in 1826. None of this material cross-referenced the Dublin examination problems or the Ladies' Diary question from Rutherford.

  William Mason's Prize Question in the Lady's and Gentleman's Diary for 1864 stands as perhaps the most extended treatment in this tradition. The solutions and commentary published the following year ran to roughly fifteen pages. By that point the almanac was near the end of its long run, but problems of this type continued in the Educational Times into the early 1900s.

  When Ross Honsberger proposed what he believed was a novel result in Mathematical Gems in 1973, he was, unknowingly, recapitulating part of this informal but vast literature.

• 5. Dublin 1820

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  In October 1820, candidates at the University of Dublin sat for the gold medal in the General Examination. The geometry paper, set on the second morning, included three consecutive problems that together state and explore Napoleon's theorem.

  Question 10 asked about equilateral triangles placed outwardly on a given triangle, with the lines joining the centers forming an equilateral triangle. Question 11 repeated the construction inwardly. Question 12 asked candidates to investigate the relation between the area of the original triangle and the areas of the two resulting equilateral triangles.

  Those problems were collected and published in Dublin Problems, a volume covering the gold medal examinations from 1816 to 1822, issued by G. and W. B. Whittaker in London in 1823. James Thomson included the result with proof in his Euclid textbook by 1834, noting in an endnote that he had encountered it only in the Dublin Problems, where it appeared without demonstration. In the second edition of 1837, Thomson extended that note with an outline of a proof by Adam D. Glasgow of Belfast, a former student Thomson described as having great taste and talent for mathematical pursuits.

  Thomson was unaware of the Ladies' Diary question from 1825 or the Gentleman's Diary appearance from 1829. J. S. Mackay later remained unaware of the Gentleman's Diary entry while noting the Ladies' Diary one. R. C. Archibald, writing in the American Mathematical Monthly for January 1920, pointed readers to the Gentleman's Diary question but had not accounted for the first published solution in the Ladies' Diary for 1826. Each researcher, working independently, found a different piece of the record.

• 6. Generalizations Beyond Triangles

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  Napoleon's theorem turns out to be a special case of broader results about polygons and their centers.

  The Petr-Douglas-Neumann theorem extends the construction to arbitrary n-gons. Erect isosceles triangles with specified apex angles on the sides of any n-gon, form a new n-gon from the free apices, then repeat the process with a different value, continuing until all values have been used in any order. The result is always a regular n-gon whose centroid coincides with the centroid of the original polygon.

  The Napoleon-Barlotti theorem gives a condition that works in both directions: the centers of regular n-gons built on the sides of an n-gon P form a regular n-gon if and only if the original polygon is an affine image of a regular n-gon.

  The Jha-Savarn generalization works with hexagons. Build equilateral triangles on the sides of a hexagon, label the apexes, and take the centroids of specific triangles formed from those apexes. Those centroids form an equilateral triangle.

  Dao Than Oai's generalization goes further still, working with a hexagon ABCDEF and equilateral triangles built on alternating sides. Two separate sets of centroids are computed, each set producing its own equilateral triangle. When the hexagon is collapsed by letting pairs of points coincide, Dao Than Oai's result reduces back to Napoleon's original theorem, which ties the centuries of generalization to the simple construction that Rutherford posed to readers of the Ladies' Diary in 1825.