The Yff circles are the two triplets of congruent circles in which each circle is tangent
to two sides of a reference triangle. In each
case, the triplets intersect pairwise in a single point. Denoting the radius 
, the exact trilinear
coordinates of the centers of the 
-, 
-,
and 
-circles are given by
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
Denoting the mutual point of intersection by 
and equating the distances to the circle
centers to 
give the radii as
 |  |  |
(4)
|
 |  |  |
(5)
|
The points through which each set of triples pass have triangle center functions
 |  |  |
(6)
|
 |  |  |
(7)
|
which are the internal similitude center and external similitude center, respectively,
of the circumcircle and incircle,
corresponding to Kimberling centers 
and 
.
Since the three circles have equal radii and intersect in a single point, Johnson's theorem states that the circle passing through the other three intersections has the same radius. These circles corresponding to the Yff circles are known as the Johnson-Yff circles.
