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, Đào Thanh Oai showed that Gibert's result follows from a property of rectangular hyperbolas. Namely, let and lie on one branch of a rectangular hyperbola , and and be the two points on , symmetrical about its center (antipodal points), where the tangents at are parallel to the line ,
Let and two points on the hyperbola the tangents at which intersect at a point on the line . If the line intersects at , and the perpendicular bisector of intersects the hyperbola at and , then the six points lie on a circle.
To get Lester's theorem from this result, take as the Kiepert hyperbola of the triangle, take to be its Fermat points, be the inner and outer Vecten points, be the orthocenter and the centroid of the triangle.
- B. Gibert (2000): [ Message 1270]. Entry in the Hyacinthos online forum, 2000-08-22. Accessed on 2014-10-09.
- Paul Yiu (2010), The circles of Lester, Evans, Parry, and their generalizations. Forum Geometricorum, volume 10, pages 175–209. MR 2868943
- Đào Thanh Oai (2014), A Simple Proof of Gibert’s Generalization of the Lester Circle Theorem Forum Geometricorum, volume 14, pages 201–202. MR 3208157
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- The Lester Circle Details of its discovery.
- Lester Circle at MathWorld
- Center of the Pohoata-Dao–Moses circles X(5607) and X(5608)
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