The Yiu 
-circle
of a reference triangle 
is the circle passing through vertex 
and the reflections of vertices 
and 
with respect to the opposite sides. The Yiu 
- and 
-circles
are then analogously defined.
The 
-circle has center
 |
(1)
|
which can also be written
 |
(2)
|
(P. Moses, pers. comm., Jan. 31, 2005).
Its 
-radius is
 |  |  |
(3)
|
 |  |  |
(4)
|
where 
,
, 
, and 
are Conway triangle
notation, 
is the circumcenter, and 
is the orthocenter (P. Moses,
pers. comm., Jan. 31, 2005).
The Yiu circles powers with respect to the vertices are
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
The Yiu circles mutually intersect in a single point, which is therefore their radical center. It has center function
 |
(10)
|
which is Kimberling center 
(the inverse in the circumcircle of the Kosnita
point 
).
The Yiu circles do not have a radical circle.
