But more is true. In any triangle ABC the centres G, H and O, respectively the
centroid, orthocentre, and circumcentre, themselves lie along one line, called the
‘Euler line’. In the case of an equilateral triangle (all sides of equal length) these
three points coincide and the resulting point is unambiguously the centre of the
triangle.
The Euler lineNapoleon’s theorem
For any triangle ABC, equilateral triangles can be constructed on each side and
from their centres a new triangle DEF is constructed. Napoleon’s theorem asserts
that for any triangle ABC, the triangle DEF is an equilateral triangle.
Napoleon’s theorem appeared in print in an English journal in 1825 a few
years after his death on St Helena in 1821. Napoleon’s ability in mathematics in
school no doubt helped him gain entrance to the artillery school, and later he got
to know the leading mathematicians in Paris when he was Emperor.
Unfortunately there is no evidence to take us further and ‘Napoleon’s theorem’ is,
like many other mathematical results, ascribed to a person who had little to do
with its discovery or its proof. Indeed, it is a theorem which is frequently being
rediscovered and extended.