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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 H and G lie on one branch of a rectangular hyperbola, and let F_+ and F_- 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 HG. Let K_+ and K_- be two points on the hyperbola where the tangents 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_+, and G_- lie on a circle. When the rectangular hyperbola is the Kiepert hyperbola and F_+ and F_- 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 P be a point on the Neuberg cubic, and let P_A be the reflection of P in the line BC, with P_B and P_C defined cyclically. The lines AP_A, BP_B, and CP_C are known to be concurrent at a point denoted as Q(P). The four points X_, X_, P, and Q(P) lie on a circle. When P is the point X(3), it is known that Q(P) = Q(X_3) = X_5, 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