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A Geometric Proof Of Napoleon’s Theorem

A Geometric Proof Of Napoleon’s Theorem. Chrissy Folsom June 8, 2000 Math 495b. Napoleon the Mathematician?. Theorem named after Napoleon Bonaparte. Questionable as to whether he really deserves credit. Some sources say he excelled in math.

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A Geometric Proof Of Napoleon’s Theorem

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  1. A Geometric Proof Of Napoleon’s 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. Napoleon’s 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. (center/centroid) Equilateral Triangles

  5. (center/centroid) Defined

  6. ABC original triangle • a = BC , b = AC , c = AB • G, I, H centroids • s = GI u t s The Setup • We will show that all three sides of GHI are equal in length.

  7. * Law of Cosines on AGI: * 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. * G Likewise for u: Proof • c is the base of an equilateral triangle, G is its centroid. Substitute for t and u in *

  9. * * Substitute

  10. Recall: * Plug it into * : * * Cosines

  11. (1) (2) c h b Look at ABC • Law of Cosines on ABC • Area of ABC: Plug in (1) and (2) to *

  12. * (1) (2) * * Plug it in • Plugging in:

  13. * * This is symmetric in a,b,c So... 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 n=3).

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