Lester's theorem

From Wikipedia, the free encyclopedia
Jump to: navigation, search
The Fermat points X_{13},X_{14}, the center X_5 of the nine-point circle (light blue), and the circumcenter X_3 of the green triangle lie on the Lester circle (black).

In Euclidean plane geometry, Lester's theorem, named after June Lester, states that in any scalene triangle, the two Fermat points, the nine-point center, and the circumcenter lie on the same circle.

Proofs[edit]

Gibert's proof using the Kiepert hyperbola[edit]

Lester's circle theorem follows from a more general result by B. Gibert (2000); namely, that every circle whose diameter is a chord of the Kiepert hyperbola of the triangle and is perpendicular to its Euler line passes through the Fermat points.[1][2]

Dao's lemma on the rectangular hyperbola[edit]

Dao's theorem on a rectangular hyperbola

In 2014, Đào Thanh Oai showed that Gibert's result follows from a property of rectangular hyperbolas. Namely, let H and G lie on one branch of a rectangular hyperbola S, and F_+ and F_- be the two points on S, symmetrical about its center (antipodal points), where the tangents at S are parallel to the line HG,

Let K_+ and K_- two points on the hyperbola the tangents at which intersect at a point E on the line HG. If the line K_+K_- intersects HG at D, and the perpendicular bisector of DE intersects the hyperbola at G_+ and G_-, then the six points F_+,F_-,E,F,G_+,G_- lie on a circle.[3]

To get Lester's theorem from this result, take S as the Kiepert hyperbola of the triangle, take F_+,F_- to be its Fermat points, K_+,K_- be the inner and outer Vecten points, H,G be the orthocenter and the centroid of the triangle.[3]

See also[edit]

References[edit]

  1. ^ B. Gibert (2000): [ Message 1270]. Entry in the Hyacinthos online forum, 2000-08-22. Accessed on 2014-10-09.
  2. ^ Paul Yiu (2010), The circles of Lester, Evans, Parry, and their generalizations. Forum Geometricorum, volume 10, pages 175–209. MR 2868943
  3. ^ a b Đào Thanh Oai (2014), A Simple Proof of Gibert’s Generalization of the Lester Circle Theorem Forum Geometricorum, volume 14, pages 201–202. MR 3208157
  • Clark Kimberling, "Lester Circle", Mathematics Teacher, volume 89, number 26, 1996.
  • June A. Lester, "Triangles III: Complex triangle functions", Aequationes Mathematicae, volume 53, pages 4–35, 1997.
  • Michael Trott, "Applying GroebnerBasis to Three Problems in Geometry", Mathematica in Education and Research, volume 6, pages 15–28, 1997.
  • Ron Shail, "A proof of Lester's Theorem", Mathematical Gazette, volume 85, pages 225–232, 2001.
  • John Rigby, "A simple proof of Lester's theorem", Mathematical Gazette, volume 87, pages 444–452, 2003.
  • J.A. Scott, "On the Lester circle and the Archimedean triangle", Mathematical Gazette, volume 89, pages 498–500, 2005.
  • Michael Duff, "A short projective proof of Lester's theorem", Mathematical Gazette, volume 89, pages 505–506, 2005.
  • Stan Dolan, "Man versus Computer", Mathematical Gazette, volume 91, pages 469–480, 2007.

External links[edit]