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The Fermat points , the center of the nine-point circle (light blue), and the circumcenter 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] The center of the Lester circle is also a triangle center. It is the center designated as X(1116) in the Encyclopedia of Triangle Centers.[9] Recently, Peter Moses discovered 21 other triangle centers lie on the Lester circle. The points are numbered X(15535) – X(15555) in the Encyclopedia of Triangle Centers.[10]

Gibert’s generalization

In 2000, Bernard Gibert proposed a generalization of the Lester Theorem involving the Kiepert hyperbola of a triangle. His result can be stated as follows: Every circle with a diameter that is a chord of the Kiepert hyperbola and perpendicular to the triangle’s Euler line passes through the Fermat points.[11]

See also

References

  1. ^ Lester, June A. (1997), “Triangles. III. Complex triangle functions”, Aequationes Mathematicae, 53 (1–2): 4–35, doi:10.1007/BF02215963, MR 1436263, S2CID 119667124
  2. ^ Kimberling, Clark (1996), “Lester circle”, The Mathematics Teacher, 89 (1): 26, JSTOR 27969621
  3. ^ Shail, Ron (2001), “A proof of Lester’s theorem”, The Mathematical Gazette, 85 (503): 226–232, doi:10.2307/3622007, JSTOR 3622007, S2CID 125392368
  4. ^ Rigby, John (2003), “A simple proof of Lester’s theorem”, The Mathematical Gazette, 87 (510): 444–452, doi:10.1017/S0025557200173620, JSTOR 3621279, S2CID 125214460
  5. ^ Scott, J. A. (2003), “Two more proofs of Lester’s theorem”, The Mathematical Gazette, 87 (510): 553–566, doi:10.1017/S0025557200173917, JSTOR 3621308, S2CID 125997675
  6. ^ Duff, Michael (2005), “A short projective proof of Lester’s theorem”, The Mathematical Gazette, 89 (516): 505–506, doi:10.1017/S0025557200178581, S2CID 125894605
  7. ^ Dolan, Stan (2007), “Man versus computer”, The Mathematical Gazette, 91 (522): 469–480, doi:10.1017/S0025557200182117, JSTOR 40378420, S2CID 126161757
  8. ^ Trott, Michael (1997), “Applying GroebnerBasis to three problems in geometry”, Mathematica in Education and Research, 6 (1): 15–28
  9. ^ Clark Kimberling, X(1116) = CENTER OF THE LESTER CIRCLE in Encyclopedia of Triangle Centers
  10. ^ Peter Moses, Preamble before X(15535) in Encyclopedia of Triangle Centers
  11. ^ Yiu, Paul. “The circles of Lester, Evans, Parry, and their generalizations” (PDF). Forum Geometricorum. 10: 175–209. Archived from the original (PDF) on 2021-10-07.