The Dao-Parry circle associated with a rectangular circumhyperbola 
of a scalene
triangle 
is the circle through the five points 
, 
, 
, 
, and the focus 
of the Kiepert
parabola in the following construction. Dao Thanh Oai (2016) proposed the configuration
as a generalization of the Parry circle, and Lozada
(2021) introduced the name.
Let 
be the isogonal conjugate of a nondegenerate
, and let the tangent line
to 
at the orthocenter 
meet 
at a finite point 
. Write 
for the center of 
and suppose 
. Assume that the line 
meets 
at two distinct finite real points 
and 
. Let 
, 
, and 
be the isogonal conjugates
of 
, 
, and 
, respectively. Assume that all three isogonal
conjugates are defined and finite, with 
different from the circumcenter 
. Let 
be the image of 
under inversion in the circumcircle.
Then the five points are concyclic
and determine the Dao-Parry circle (Dao Thanh Oai 2016, Lozada 2021).
The point 
lies on the Jerabek hyperbola,
so its isogonal conjugate 
lies on the Euler line. Its
inverse point 
also lies on the Euler line,
and the Dao-Parry circle is orthogonal to the
circumcircle. Its center lies on the line
through 
and the center 
of the Parry circle (Lozada 2021).
Taking 
to be the Kiepert hyperbola gives the Parry
circle (Dao Thanh Oai 2016, Lozada 2021).
