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 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

lie on one branch of a rectangular hyperbola, and let

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

be two points on the hyperbola where the tangents intersect at a point

, and the perpendicular bisector of

intersects the hyperbola at

lie on a circle.

When the rectangular hyperbola is the Kiepert hyperbola 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.

be a point on the Neuberg cubic, and let

defined cyclically.

are known to be concurrent at a point denoted as

lie on a circle.

, making Dao's generalization a restatement of the Lester Theorem.