The Fermat points X 13 , X 14 {\displaystyle X_{13},X_{14}}, the center X 5 {\displaystyle X_{5}} of the nine-point circle (light blue), and the circumcenter X 3 {\displaystyle X_{3}} of the green triangle lie on the Lester circle (black).

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, and the circle through these points was called the Lester circle by Clark Kimberling. Lester proved the result by using the properties of complex numbers; 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 centers lie on the Lester circle. The points are numbered X(15535) – X(15555) in the Encyclopedia 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.

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 {\displaystyle H} and G {\displaystyle G} lie on one branch of a rectangular hyperbola, and let F + {\displaystyle F_{+}} and F − {\displaystyle 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 H G {\displaystyle HG}. Let K + {\displaystyle K_{+}} and K − {\displaystyle K_{-}} be two points on the hyperbola where the tangents intersect at a point E {\displaystyle E} on the line H G {\displaystyle HG}. If the line K + K − {\displaystyle K_{+}K_{-}} intersects H G {\displaystyle HG} at D {\displaystyle D}, and the perpendicular bisector of D E {\displaystyle DE} intersects the hyperbola at G + {\displaystyle G_{+}} and G − {\displaystyle G_{-}}, then the six points F + {\displaystyle F_{+}}, F − , {\displaystyle F_{-},} E , {\displaystyle E,} F , {\displaystyle F,} G + {\displaystyle G_{+}}, and G − {\displaystyle G_{-}} lie on a circle. When the rectangular hyperbola is the Kiepert hyperbola and F + {\displaystyle F_{+}} and F − {\displaystyle F_{-}} are the two Fermat points, Dao's generalization becomes Gibert's generalization.

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 {\displaystyle P} be a point on the Neuberg cubic, and let P A {\displaystyle P_{A}} be the reflection of P {\displaystyle P} in the line B C {\displaystyle BC}, with P B {\displaystyle P_{B}} and P C {\displaystyle P_{C}} defined cyclically. The lines A P A {\displaystyle AP_{A}}, B P B {\displaystyle BP_{B}}, and C P C {\displaystyle CP_{C}} are known to be concurrent at a point denoted as Q ( P ) {\displaystyle Q(P)}. The four points X 13 {\displaystyle X_{13}}, X 14 {\displaystyle X_{14}}, P {\displaystyle P}, and Q ( P ) {\displaystyle Q(P)} lie on a circle. When P {\displaystyle P} is the point X ( 3 ) {\displaystyle X(3)}, it is known that Q ( P ) = Q ( X 3 ) = X 5 {\displaystyle Q(P)=Q(X_{3})=X_{5}}, making Dao's generalization a restatement of the Lester Theorem.

See also

External links