napoleon's theorem

Input interpretation

Napoleon's theorem

Illustration

Illustration

Definition

If equilateral triangles Δ A B E_(A B), Δ B C E_(B C), and Δ A C E_(A C) are erected externally on the sides of any triangle Δ A B C, then their centers N_(A B), N_(B C), and N_(A C), respectively, form an equilateral triangle (the outer Napoleon triangle) Δ N_(A B) N_(B C) N_(A C). An additional property of the externally erected triangles also attributed to Napoleon is that their circumcircles concur in the first Fermat point X. Furthermore, the lines A E_(B C), B E_(A C), and C E_(A B) connecting the vertices of Δ A B C with the opposite vectors of the erected triangles also concur at X.
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Related terms

equilateral triangle | Fermat points | inner Napoleon triangle | Kiepert hyperbola | Napoleon points | outer Napoleon triangle | Petr-Neumann-Douglas theorem | similar | van Aubel's theorem

Subject classifications

MathWorld

triangle properties

MSC 2010

51M04

Associated person

Napoleon
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