Napoleon’s theorem


If equilateral trianglesMathworldPlanetmath are erected externally on the three sides of any given triangleMathworldPlanetmath, then their centres are the vertices of an equilateral triangle.

If we embed the statement in the complex plane, the proof is a mere calculation. In the notation of the figure, we can assume that A=0, B=1, and C is in the upper half plane. The hypotheses are

1-0Z-0=C-1X-1=0-CY-C=α (1)

where α=expπi/3, and the conclusionMathworldPlanetmath we want is

N-LM-L=α (2)


L=1+X+C3  M=C+Y+03  N=0+1+Z3.

From (1) and the relationMathworldPlanetmath α2=α-1, we get X,Y,Z:


and so

3(M-L) = Y-1-X
= (2α-1)C-1-α
3(N-L) = Z-X-C
= (α-2)C+1-2α
= (2α-2-α)C-α+1-α
= (2α2-α)C-α-α2
= 3(M-L)α

proving (2).

Remarks: The attribution to Napoléon Bonaparte (1769-1821) is traditional, but dubious. For more on the story, see

Title Napoleon’s theorem
Canonical name NapoleonsTheorem
Date of creation 2013-03-22 13:48:50
Last modified on 2013-03-22 13:48:50
Owner drini (3)
Last modified by drini (3)
Numerical id 7
Author drini (3)
Entry type Theorem
Classification msc 51M04