A Geometric Proof Of Napoleon

1
A Geometric Proof Of Napoleons Theorem

• Chrissy Folsom
• June 8, 2000
• Math 495b

2
Napoleon the Mathematician?

• Theorem named after Napoleon Bonaparte

• Questionable as to whether he really deserves
  credit.
• Some sources say he excelled in math

• Earliest definite appearance of theorem 1825 by
  Dr. W. Rutherford in The Ladies Diary

3
Napoleons Theorem

• Given any triangle, construct an equilateral
  triangle on each of its legs. Then the centers
  of the three outer triangles form another
  equilateral triangle (Napoleon triangle).

4
Equilateral Triangles
5
Defined
6
The Setup

• We will show that all three sides of GHI are
  equal in length.

7
Proof (Find s in terms of a,b,c)
(A both point and angle)
Question Can we find t in terms of c?
YES!!!
8
Proof

• c is the base of an equilateral triangle, G is
  its centroid.

Substitute for t and u in
9
Substitute
10
Cosines
11
Look at ABC

• Law of Cosines on ABC

• Area of ABC

Plug in (1) and (2) to
12
Plug it in

• Plugging in

13
Are We Done?
Hence, an equilateral triangle.
Yes, We Are Done (with the proof)!!!
14
More Neat Stuff Tiling
(1) Rotate original triangle 120o about centroid
of each adjacent equilateral
(2) Connect exposed vertices to get equilateral
triangles
(3) Connect vertices of 3 new equilateral
triangles
(4) Another equilateral triangle!!
15
Conclusions

• Some Generalizations
• If similar triangles of any shape are added onto
  the original triangle, then any triple of
  corresponding points on triangles forms a
  triangle of same shape.
• Begin with arbitrary n-gon. Attach a regular
  n-gon to each side. Connect similar points and
  get another regular n-gon. (Napoleon when n3).