In general, the internal similitude center of two circles 
and 
with centers given in Cartesian coordinates is
given by
 |
(1)
|
In trilinear coordinates, the internal center of similitude is given by 
, where
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
The incircle and circumcircle of a triangle 
have two similitude
centers, namely the internal center of similitude Si and the external
similitude center Se. The internal center of similitude of these two circles
Si is the isogonal conjugate of the
Gergonne point of 
. It is Kimberling
center 
and has equivalent triangle center functions
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
The two points Si and Se share certain similar properties, but there seems to be no straightforward analogy between the two. For instance, Si is
the homothetic center of the tangential,
intangents, and extangents
triangles of triangle 
taken pairwise, but the only comparable property of
the external similitude center Se
is more complicated: Se is the homothetic
center of the tangential triangle and
the reflection of the intangents triangle
in the incenter of 
.
The following table summarizes the internal similitude centers for a number of named circles.
| circle 1 | circle 2 | Kimberling | internal similitude center |
| Adams circle | Conway circle |  | incenter |
| Adams circle | incircle |  | incenter |