Transcript Slide 1

~is said to be one of the most-often rediscovered
results in mathematics
~earliest
appearance of this
theorem was said to
be is an 1825 article
by Dr. W. Rutherford
in "The Ladies
Diary“. Although he
was probably not
the first discoverer.
~The theorem is named after Napoleon
Bonaparte, although he was not the first to
discover it but apparently found it
independently and so it bears his name.
http://www.mathpages.com/home/kmath270/kmath270.htm
If we construct equilateral triangles on the sides of any
triangle - all outward or all inward - the centers of those
equilateral triangles themselves form an equilateral
triangle.
http://www.mathpages.com/HOME/kmath270/kmath270.htm
Napoleon's Theorem is very similar to that
of Van Aubel’s theorem which states that:
Given a quadrilateral, place a square
outwardly on each side, and connect the
centers of opposite squares. Then the two
lines are of equal length and cross at a
right angle.
k = 9.34 cm
l = 9.34 cm
B
A
O
k
mAOB = 90.00
mBOC = 90.00
mCOD = 90.00
mDOA = 90.00
http://mathworld.wolfram.com/vanAubelsTheorem.html
D
l
C
Triangles
out side
B
E
H
G
D
C
A
m GH = 5.07 cm
m IG = 5.07 cm
m HI = 5.07 cm
I
F
B
Triangles Inside
E
F
D
C
A
m DE = 1.41 cm
m EF = 1.41 cm
m FD = 1.41 cm
proof:
Since IAC = GAB = 30, w e can apply the Law of Cosines to compute the length of s ide s:
s2 = u2 + t2 - 2ut·cos(A + 60
)
Since the centroid of an equilateral triangle lies on the median, 2/3 of the distanc e from the vertex
to the midpoint of the opposite side, w e have:
t = (2/3) * 3 /2 * c = c/ 3
F
u = (2/3) * 3 /2 * b = b/ 3
and then becomes :
3 * s2 = b2 + c2 - 2bc * c os(A +60
)
A
Expanding the cos ine of the sum, w e get:
cos (A + 60) = cos (A)/2 - sin(A) *3 /2
Then, using substitution, w e get:
3 * s2 = b2 + c2 - bc * cos (A) - bc * sin(A) *3 /2
Now , w e apply the Law of Cosines toABC:
a2 = b2 + c2 - 2bc·cos(A)
u
I
E
G
ts
C
B
H
and recall (the derivation of the Law of Sines):
2·Area(A BC) = bc·sin(A)
and by us ing subs titution, w e get:
3 * s2 = (1/2)(a2 + b2 + c2) + 2 * 3 * Area( ABC)
D
Since this last equation is symmetrical in a,b, and c, it follow s that the triangle c onnecting the three
centroids is equilateral
http://www.cut-the-knot.org/proofs/napoleon.shtml
History:
http://www.mathpages.com/home/kmath270/kmath270.htm
Van Aubel’s Theorem:
http://mathworld.wolfram.com/vanAubelsTheorem.html
Napoleon’s Theorem:
http://www.mathpages.com/HOME/kmath270/kmath270.htm
Proof of Napoleon’s Theorem:
http://www.cut-the-knot.org/proofs/napoleon.shtml