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, Dao 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

A conjectured 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 $AP_A$ , $BP_B$ , $CP_C$ are concurrent. Let $Q(P)$ be the point of concurrency. 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.

Notes

↑ Paul Yiu (2010), The circles of Lester, Evans, Parry, and their generalizations. Forum Geometricorum, volume 10, pages 175–209. MR 2868943
↑ B. Gibert (2000): [ Message 1270]. Entry in the Hyacinthos online forum, 2000-08-22. Accessed on 2014-10-09.
↑ Dao Thanh Oai (2014), A Simple Proof of Gibert’s Generalization of the Lester Circle Theorem Forum Geometricorum, volume 14, pages 201–202. MR 3208157
↑ "X(7668) = POLE OF X(115)X(125) WITH RESPECT TO THE NINE-POINT CIRCLE". 2015-06-01.

References

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

The Lester Circle Details of its discovery.
Lester Circle at MathWorld
Center of the Pohoata-Dao–Moses circles X(5607) and X(5608)

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