
In
Euclidean plane
In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, Lester's theorem states that in any
scalene triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensional ...
, 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, and the circle through these points was called the Lester circle by
Clark Kimberling.
Lester proved the result by using the properties of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s; subsequent authors have given elementary proofs, proofs using vector arithmetic, and computerized proofs. 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. Recently, Peter Moses discovered 21 other
triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
centers lie on the Lester circle. The points are numbered X(15535) – X(15555) in the
Encyclopedia of Triangle Centers.
Peter Moses, Preamble before X(15535) in
ncyclopedia of Triangle Centers
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. [Paul Yiu, ''The circles of Lester, Evans, Parry, and their generalizations'', Forum Geometricorum, volume 10, pages 175–209]
[Dao Thanh Oai, ''A Simple Proof of Gibert’s Generalization of the Lester Circle Theorem'', Forum Geometricorum, volume 14, pages 201–202]
Dao's generalizations
Dao's first generalization
In 2014, Dao Thanh Oai extended Gibert's result to every rectangular hyperbola. The generalization is as follows: Let and lie on one branch of a rectangular hyperbola, and let and be the two points on the hyperbola that are symmetrical about its center ( antipodal points), where the tangents at these points are parallel to the line . Let and be two points on the hyperbola where the tangents intersect at a point on the line . If the line intersects at , and the perpendicular bisector of intersects the hyperbola at and , then the six points , , and lie on a circle. When the rectangular hyperbola is the Kiepert hyperbola and and are the two Fermat points, Dao's generalization becomes Gibert's generalization. [Ngo Quang Duong, ''Generalization of the Lester circle'', Global Journal of Advanced Research on Classical and Modern Geometries, Vol.10, (2021), Issue 1, pages 49–61]
Dao's second generalization
In 2015, Dao Thanh Oai proposed another generalization of the Lester circle, this time associated with the Neuberg cubic. It can be stated as follows: Let be a point on the Neuberg cubic, and let be the reflection of in the line , with and defined cyclically. The lines , , and are known to be concurrent at a point denoted as . The four points , , , and lie on a circle. When is the point , it is known that , making Dao's generalization a restatement of the Lester Theorem. [Dao Thanh Oai, ''Generalizations of some famous classical Euclidean geometry theorems'', International Journal of Computer Discovered Mathematics, Vol.1, (2016), Issue 3, pages 13–20]
ncyclopedia of Triangle Centers[César Eliud Lozada, Preamble before X(42740) in ]
ncyclopedia of Triangle Centers
See also
* Parry circle
*
* van Lamoen circle
References
External links
*{{mathworld, id=LesterCircle, title=Lester Circle
Theorems about triangles and circles