Lester's theorem

Gibert's proof using the Kiepert hyperbola

Lester's circle theorem follows from a more general result by B. Gibert (2000); namely, that every circle whose diameter is a chord of the Kiepert hyperbola of the triangle and is perpendicular to its Euler line passes through the Fermat points.

Dao's lemma on the rectangular hyperbola

In 2014, Dao Thanh Oai showed that Gibert's result follows from a property of rectangular hyperbolas. Namely, let H and G lie on one branch of a rectangular hyperbola S , and F + and F − be the two points on S , symmetrical about its center (antipodal points), where the tangents at S are parallel to the line H G ,

Let K + and K − two points on the hyperbola the tangents at which intersect at a point E on the line H G . If the line K + K − intersects H G at D , and the perpendicular bisector of D E intersects the hyperbola at G + and G − , then the six points F + , F − , E , F , G + , G − lie on a circle.

To get Lester's theorem from this result, take S as the Kiepert hyperbola of the triangle, take F + , F − to be its Fermat points, K + , K − be the inner and outer Vecten points, H , G be the orthocenter and the centroid of the triangle.

Generalisation

A conjectured generalization of the Lester theorem was published in Encyclopedia of Triangle Centers as follows: Let P be a point on the Neuberg cubic. Let P A be the reflection of P in line B C , and define P B and P C cyclically. It is known that the lines A P A , B P B , C P C are concurrent. Let Q ( P ) be the point of concurrency. Then the following 4 points lie on a circle: X 13 , X 14 , P , Q ( P ) . When P = X ( 3 ) , it is well-known that Q ( P ) = Q ( X 3 ) = X 5 , the conjecture becomes Lester theorem.