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 from 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]
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 from 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]