Lester's theorem

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]

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. ^[11]^[12]

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. ^[12]^[13]

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 $X_{13}$ , $X_{14}$ , $P$ , 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. ^[13]^[14]^[15]^[16]