The Dao conjugate is an isoconjugation defined by a circumconic construction (Lozada 2021). Varying
the circumconic gives a family of conjugations. Lozada
(2021) catalogs particular Dao conjugates of triangle
centers as 
through 
.
To construct the Dao conjugate, let 
be a circumconic of a
reference triangle 
that is a central conic,
and let its center 
lie on none of the side lines. Given
a point 
on none of these side lines, let
, 
, and 
meet 
again at 
, 
, and 
, respectively. The lines through
, 
, and 
parallel to 
, 
,
and 
meet 
again at 
,
, and 
, respectively. Assume that these second intersections
are finite and distinct from the corresponding starting points,
and that 
,
, and 
are well-defined lines. Then
these three lines are concurrent.
Their common point 
, possibly a point at infinity,
is the 
-Dao
conjugate of 
(Lozada 2021).
If 
and 
are in barycentric
coordinates, then
 |
For fixed 
,
the coordinate formula defines an involution on the
points of the projective
plane with 
,
 |
For the circumcircle, the construction gives the isogonal conjugate of 
. For the Steiner circumellipse,
it gives the isotomic conjugate of 
(Lozada 2021).
