Proving this theorem was a class assignment. My proof may or may not be original (probably not).
Theorem (
BONAPARTE)
Let
ABC be a triangle.
Erect
equilateral triangles A'BC,
AB'C,
ABC' so either all overlap with Δ
ABC or none overlap, with
centroids a,
b,
c respectively. Then Δ
abc is an
equilateral triangle (known as Napoleon's triangle).
_.B'
_ ./ 
_ ./ /
_ ./ 
_ ./ /
C'________________A / 
\ /\ .b /
\ / \ 
\ .c / \ /
\ / \ 
\ / \ /
\ / \ 
\ / \ /
\ / \ 
\/________________________\/
B \ /C
\ /
\ /
\ /
\ .a /
\ /
\ /
\ /
\ /
\ /
\ /
\ /
\/
A'
Proof
Make two comparisons between Δ
cAb and Δ
BAB' to see that they are
similar triangles:
∠ cAb = ∠ cAB + ∠ BAC + ∠ CAb
= π/6 + ∠ BAC + π/6
= ∠ BAC + π/3
∠ BAB' = ∠ BAC + ∠ CAB'
= ∠ BAC + π/3
(same angles)
By the fixed ratios of lengths in an equilateral triangle:
c  A = B  A / sqrt(3)
b  A = B'  A / sqrt(3)
∴
c  A / b  A = B  A / B'  A
(preserved ratio of side lengths)
They are similar triangles. Hence the third length must have the same ratio:
c  b = B'  B / sqrt(3)
By reasons of symmetry,
a  b = B'  B / sqrt(3)
Hence

a  b = 
b  c
and by reasons of symmetry, all sides of
Δ
abc are of equal length.
This proof works for the case where the constructed equilateral triangles point inward, too.
QED
Napoleon Points
Lines
l_{aA} ,
l_{bB} , and
l_{cC} meet at the first Napoleon point when the three equilateral triangles point away from Δ
ABC. When they point toward Δ
ABC, the lines meet at a different point known as the second Napoleon point.
Fermat Point
The point
P of a triangle
ABC that minimizes the sum of distances 
A  P + 
B  P + 
C  P is known as the Fermat point.
When each of the three angles of Δ
ABC are less than 120°, the Fermat point is the
concurrent point of lines
l_{A'A},
l_{B'B}, and
l_{C'C}. It is also where the
circumcircles of the erected
equilateral triangles meet. When one of the
vertices has an angle greater than 120
°, then that
vertex is the Fermat point.
The Fermat point was not discovered by Fermat himself. Instead, he was the one who asked where such a point would exist, to which Toricelli was the first to answer this question.