Napoleon’s theorem
Theorem.
If equilateral triangles
are erected externally on the three sides of any given triangle
, 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 , , and is in the upper half plane. The hypotheses are
(1) |
where , and the conclusion we want is
(2) |
where
From (1) and the relation , we get :
and so
proving (2).
Remarks: The attribution to Napoléon Bonaparte (1769-1821) is traditional, but dubious. For more on the story, see http://www.mathpages.com/home/kmath270/kmath270.htmMathPages.
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 |