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, 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 from 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 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.

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.