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In mathematics, 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 Napoleon triangle (inner and outer). 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). However, it may just date back to W. Rutherford's 1825 publication The Ladies' Diary, four years after the French emperor's death.[1]

Proofs

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 an homothety of ratio √3 with the same center, and that LN also becomes CZ after a counterclockwise rotation of 30° around B and an homothety of ratio √3 with the same center. The respective spiral similarities[2] are A(√3,-30°) and B(√3,30°). That implies MN = LN and the angle between them must be 60°.[3]

Analytically, it can be determined[4] that each of the three segments of the LMN triangle has a length of:

$$\sqrt{{a^2+b^2+c^2 \over 6} + {\sqrt{(a+b+c)(a+b-c)(a-b+c)(-a+b+c)} \over {2\sqrt{3}}}}$$

There are in fact many proofs of the theorem's statement, including a trigonometric one,[4] a symmetry-based approach,[5] and proofs using complex numbers.[4]
Background
Extract from the 1826 Ladies' Diary giving geometric and analytic proofs

The following entry appeared on page 47 in the Ladies' Diary of 1825. As the earliest known reference it may fairly be regarded as the official birth certificate of Napoléon's theorem.

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 clearly a very able 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.

Plainly there is no reference to Napoléon in either the question or the published responses, though the Editor evidently omitted some submissions. Also Rutherford himself does not appear amongst the named solvers.

Several intriguing mysteries survive to this day :-

did Rutherford discover the theorem or was it communicated to him by someone else?
when and by whom was the theorem first attributed to Napoléon?
did Napoléon have anything to do with the initial discovery or proof of the theorem, and if not why does it bear his name?

Napoleon's problem

Notes

^ http://mathworld.wolfram.com/NapoleonsTheorem.html
^ Weisstein, Eric W., "Spiral Similarity" from MathWorld.
^ For a visual demonstration see Napoleon's Theorem via Two Rotations at Cut-the-Knot.
^ a b c "Napoleon's Theorem" at MathPages.com.
^ Proof #2 (an argument by symmetrization)

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" from 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.

Mathematics Encyclopedia