Lester's theorem (original) (raw)

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Several points associated with a scalene triangle lie on the same circle

The Fermat points X 13 , X 14 {\displaystyle X_{13},X_{14}} $X_{13},X_{14}$ , the center X 5 {\displaystyle X_{5}} $X_{5}$ of the nine-point circle (light blue), and the circumcenter X 3 {\displaystyle X_{3}} $X_{3}$ 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,[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]

Gibert's generalization

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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]

Dao's generalizations

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Dao's first generalization

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In 2014, Dao Thanh Oai extended Gibert's result to every rectangular hyperbola. The generalization is as follows: Let H {\displaystyle H} $H$ and G {\displaystyle G} $G$ lie on one branch of a rectangular hyperbola, and let F + {\displaystyle F_{+}} $F_{+}$ and F − {\displaystyle F_{-}} $F_{-}$ 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} $HG$ . Let K + {\displaystyle K_{+}} $K_{+}$ and K − {\displaystyle K_{-}} $K_{-}$ be two points on the hyperbola where the tangents intersect at a point E {\displaystyle E} $E$ on the line H G {\displaystyle HG} $HG$ . If the line K + K − {\displaystyle K_{+}K_{-}} $K_{+}K_{-}$ intersects H G {\displaystyle HG} $HG$ at D {\displaystyle D} $D$ , and the perpendicular bisector of D E {\displaystyle DE} $DE$ intersects the hyperbola at G + {\displaystyle G_{+}} $G_{+}$ and G − {\displaystyle G_{-}} $G_{-}$ , then the six points F + {\displaystyle F_{+}} $F_{+}$ , F − , {\displaystyle F_{-},} $F_{-},$ E , {\displaystyle E,} $E,$ F , {\displaystyle F,} $F,$ G + {\displaystyle G_{+}} $G_{+}$ , and G − {\displaystyle G_{-}} $G_{-}$ lie on a circle. When the rectangular hyperbola is the Kiepert hyperbola and F + {\displaystyle F_{+}} $F_{+}$ and F − {\displaystyle F_{-}} $F_{-}$ are the two Fermat points, Dao's generalization becomes Gibert's generalization. [12][13]

Dao's second generalization

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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} $P$ be a point on the Neuberg cubic, and let P A {\displaystyle P_{A}} $P_{A}$ be the reflection of P {\displaystyle P} $P$ in the line B C {\displaystyle BC} $BC$ , with P B {\displaystyle P_{B}} $P_{B}$ and P C {\displaystyle P_{C}} $P_{C}$ defined cyclically. The lines A P A {\displaystyle AP_{A}} $AP_{A}$ , B P B {\displaystyle BP_{B}} $BP_{B}$ , and C P C {\displaystyle CP_{C}} $CP_{C}$ are known to be concurrent at a point denoted as Q ( P ) {\displaystyle Q(P)} $Q(P)$ . The four points X 13 {\displaystyle X_{13}} $X_{13}$ , X 14 {\displaystyle X_{14}} $X_{14}$ , P {\displaystyle P} $P$ , and Q ( P ) {\displaystyle Q(P)} $Q(P)$ lie on a circle. When P {\displaystyle P} $P$ is the point X ( 3 ) {\displaystyle X(3)} $X(3)$ , it is known that Q ( P ) = Q ( X 3 ) = X 5 {\displaystyle Q(P)=Q(X_{3})=X_{5}} $Q(P)=Q(X_{3})=X_{5}$ , making Dao's generalization a restatement of the Lester Theorem. [13][14][15][16]