Lester's theorem

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Short description: Several points associated with a scalene triangle lie on the same circle
The Fermat points X13,X14, the center X5 of the nine-point circle (light blue), and the circumcenter X3 of the green triangle lie on the Lester circle (black).

In Euclidean plane geometry, Lester's theorem states that in any scalene triangle, the two Fermat points, the nine-point center, and the circumcenter lie on the same circle. The result is named after June Lester, who published it in 1997,[1] and the circle through these points was called the Lester circle by Clark Kimberling.[2] Lester proved the result by using the properties of complex numbers; subsequent authors have given elementary proofs[3][4][5][6], proofs using vector arithmetic,[7] and computerized proofs.[8]

See also

References

  1. ↑ Lester, June A. (1997), "Triangles. III. Complex triangle functions", Aequationes Mathematicae 53 (1–2): 4–35, doi:10.1007/BF02215963 
  2. ↑ "Lester circle", The Mathematics Teacher 89 (1): 26, 1996 
  3. ↑ Shail, Ron (2001), "A proof of Lester's theorem", The Mathematical Gazette 85 (503): 226–232, doi:10.2307/3622007 
  4. ↑ Rigby, John (2003), "A simple proof of Lester's theorem", The Mathematical Gazette 87 (510): 444–452, doi:10.1017/S0025557200173620 
  5. ↑ Scott, J. A. (2003), "Two more proofs of Lester's theorem", The Mathematical Gazette 87 (510): 553–566, doi:10.1017/S0025557200173917 
  6. ↑ Duff, Michael (2005), "A short projective proof of Lester's theorem", The Mathematical Gazette 89 (516): 505–506, doi:10.1017/S0025557200178581 
  7. ↑ Dolan, Stan (2007), "Man versus computer", The Mathematical Gazette 91 (522): 469–480, doi:10.1017/S0025557200182117 
  8. ↑ Trott, Michael (1997), "Applying GroebnerBasis to three problems in geometry", Mathematica in Education and Research 6 (1): 15–28, http://library.wolfram.com/infocenter/Articles/1754/ 




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