Lester's theorem
In Euclidean plane geometry, Lester's theorem, named after June Lester, states that in any scalene triangle, the two Fermat points, the nine-point center, and the circumcenter lie on the same circle.
Proofs
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.
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