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In Euclidean plane geometry, **Lester's theorem** states that in any scalene triangle, 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,^{[1]} and the circle through these points was called the **Lester circle** by Clark Kimberling.^{[2]}
Lester proved the result by using the properties of complex numbers; subsequent authors have given elementary proofs^{[3]}^{[4]}^{[5]}^{[6]}, proofs using vector arithmetic,^{[7]} and computerized proofs.^{[8]} 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. ^{[9]} Recently, Peter Moses discovered 21 other triangle centers lie on the **Lester circle**. The points are numbered X(15535) – X(15555) in the Encyclopedia of Triangle Centers.^{[10]}

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. ^{[11]}^{[12]}

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. ^{[12]}^{[13]}

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. ^{[13]}^{[14]}^{[15]}^{[16]}

**^**Lester, June A. (1997), "Triangles. III. Complex triangle functions",*Aequationes Mathematicae*,**53**(1–2): 4–35, doi:10.1007/BF02215963, MR 1436263, S2CID 119667124**^**Kimberling, Clark (1996), "Lester circle",*The Mathematics Teacher*,**89**(1): 26, JSTOR 27969621**^**Shail, Ron (2001), "A proof of Lester's theorem",*The Mathematical Gazette*,**85**(503): 226–232, doi:10.2307/3622007, JSTOR 3622007, S2CID 125392368**^**Rigby, John (2003), "A simple proof of Lester's theorem",*The Mathematical Gazette*,**87**(510): 444–452, doi:10.1017/S0025557200173620, JSTOR 3621279, S2CID 125214460**^**Scott, J. A. (2003), "Two more proofs of Lester's theorem",*The Mathematical Gazette*,**87**(510): 553–566, doi:10.1017/S0025557200173917, JSTOR 3621308, S2CID 125997675**^**Duff, Michael (2005), "A short projective proof of Lester's theorem",*The Mathematical Gazette*,**89**(516): 505–506, doi:10.1017/S0025557200178581, S2CID 125894605**^**Dolan, Stan (2007), "Man versus computer",*The Mathematical Gazette*,**91**(522): 469–480, doi:10.1017/S0025557200182117, JSTOR 40378420, S2CID 126161757**^**Trott, Michael (1997), "Applying GroebnerBasis to three problems in geometry",*Mathematica in Education and Research*,**6**(1): 15–28**^**Clark Kimberling, X(1116) = CENTER OF THE LESTER CIRCLE in Encyclopedia of Triangle Centers**^**Peter Moses, Preamble before X(15535) in Encyclopedia of Triangle Centers**^**Paul Yiu,*The circles of Lester, Evans, Parry, and their generalizations*, Forum Geometricorum, volume 10, pages 175–209, ISSN 1534-1178- ^
^{a}^{b}Dao Thanh Oai,*A Simple Proof of Gibert’s Generalization of the Lester Circle Theorem*, Forum Geometricorum, volume 14, pages 201–202, ISSN 1534-1178 - ^
^{a}^{b}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, ISSN 2284-5569 **^**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, ISSN 2367-7775**^**Kimberling, X(7668) = POLE OF X(115)X(125) WITH RESPECT TO THE NINE-POINT CIRCLE in Encyclopedia of Triangle Centers**^**César Eliud Lozada, Preamble before X(42740) in Encyclopedia of Triangle Centers

- Weisstein, Eric W. "Lester Circle".
*MathWorld*.