The Dao-Lester circle of a finite point 
on the Neuberg cubic of a
scalene triangle 
is the circle through 
, the two Fermat points,
and the point of concurrence 
in the construction below. Dao Thanh Oai (2016) proposed
the generalization, Ngo (2021) gave a proof, and Lozada (2021) introduced the name
Dao-Lester circle.
Let 
, 
, and 
be the reflections of 
in the sidelines 
, 
, and 
, respectively. Suppose the lines 
, 
, and 
are defined and concur at
a unique finite point 
. If three of 
, 
, and the two Fermat
points form a triangle, all four points
are concyclic and determine the Dao-Lester circle (Ngo
2021).
The center 
lies on the perpendicular bisector of the
two Fermat points, the line
through Kimberling centers 
and 
(Lozada 2021). For 
in barycentric
coordinates, write the barycentric coordinates
of 
as 
. In Conway
triangle notation, 
, where
 |
The other two coordinates follow by cyclic permutation of 
, 
, and 
together with 
, 
, and 
, leaving 
fixed (Lozada 2021).
When 
is the circumcenter 
, 
is the nine-point center 
. The construction then gives the classical Lester
circle, whose center is 
. Taking 
to be the incenter 
, the orthocenter 
, or the first
isodynamic point 
gives centers 
, 
, and 
, respectively (Lozada 2021).
