Lester's theorem

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

Gibert's proof using the Kiepert hyperbola

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

Dao's lemma 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.

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]

Generalisation

A generalization Lester circle associated with Neuberg cubic: P, Q(P), X_{13}, X_{14} lie on a circle

There is a conjecture generalization of the Lester theorem was published in Encyclopedia of Triangle Centers as follows: Let P be a point on the Neuberg cubic. Let P_A be the reflection of P in line BC, and define P_B and P_C cyclically. It is known that the lines AP_A, BP_B, CP_C concur. Let Q(P) be the point of concurrence. Then the following 4 points lie on a circle: X_{13}, X_{14}, P, Q(P). [4] When P=X(3), it is well-known that Q(P)=Q(X_3)=X_5, the conjecture becomes Lester theorem.

See also

Notes

  1. Paul Yiu (2010), The circles of Lester, Evans, Parry, and their generalizations. Forum Geometricorum, volume 10, pages 175–209. MR 2868943
  2. B. Gibert (2000): [ Message 1270]. Entry in the Hyacinthos online forum, 2000-08-22. Accessed on 2014-10-09.
  3. 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
  4. "X(7668) = POLE OF X(115)X(125) WITH RESPECT TO THE NINE-POINT CIRCLE". 2015-06-01.

References

External links

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