Napoleon's theorem

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 (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.

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

Analytically, it can be determined 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's also a trigonometric proof of the theorem's statement.

See also

• Napoleon's problem

External links

• Napoleon's Theorem and Generalizations
• To see the construction
• Napoleon's Theorem by Jay Warendorff, The Wolfram Demonstrations Project.
• Weisstein, Eric W. "Napoleon's Theorem". MathWorld.
• Weisstein, Eric W. "Spiral Similarity". MathWorld.
• Napoleon's Theorem and some generalizations, variations & converses at Dynamic Geometry Sketches
• Napoleon's Theorem, Two Simple Proofs

Napoleon's theorem at PlanetMath.