There are four completely different definitions of the so-called Apollonius circles:
1. The set of all points whose distances from two fixed points are in a constant ratio 
(Durell 1928, Ogilvy 1990).
2. One of the eight circles that is simultaneously tangent to three given circles (i.e., a circle solving Apollonius' problem for three circles).
3. One of the three circles passing through a vertex and both isodynamic points 
and 
of a triangle (Kimberling 1998, p. 68).
4. The circle that touches all three excircles of a triangle and encompasses them (Kimberling 1998, p. 102).
Given one side of a triangle and the ratio of the lengths of the other two sides, the locus of the third polygon
vertex is the Apollonius circle (of the first type) whose center
is on the extension of the given side. For a given triangle,
there are three circles of Apollonius. Denote the three Apollonius circles (of the
first type) of a triangle by 
, 
, and 
, and their centers 
, 
, and 
. The center 
is the intersection of the side 
with the tangent to the circumcircle
at 
.
is also the pole of the symmedian point 
with respect to circumcircle.
The centers 
,
,
and 
are collinear on the polar
of 
with regard to its circumcircle, called the Lemoine
axis. The circle of Apollonius 
is also the locus of a point whose pedal
triangle is isosceles such that 
.
The eight Apollonius circles of the second type are illustrated above.
Let 
and 
be points on the side line 
of a triangle 
met by the interior and exterior angle
bisectors of angles 
. Then the circle with diameter 
is called the 
-Apollonian circle. Similarly, construct the 
- and 
-Apollonian circles (Johnson 1929, pp. 294-299). The Apollonian
circles pass through the vertices 
, 
, and 
, and through the two isodynamic
points 
and 
(Kimberling 1998, p. 68). The 
-Apollonius circle has center with trilinears
 |
(1)
|
and radius
 |
(2)
|
where 
is the circumradius of the reference
triangle.
Because the Apollonius circles intersect pairwise in the isodynamic points, they share a common radical line
 |
(3)
|
which is the central line 
corresponding to Kimberling
center 
,
the isogonal conjugate of the Kiepert
parabola focus 
.
The vertices of the D-triangle lie on the respective Apollonius circles.
The circle which touches all three excircles of a triangle and encompasses them is often known as "the" Apollonius circle (Kimberling 1998, p. 102). It has circle function
 |
(4)
|
which corresponds to Kimberling center 
. Its center has triangle center
function
![alpha_(970)=a[-b^5-c^5+a^3(b+c)^2+a(ab+ac-2bc)(b^2+c^2)-bc(b^3+c^3)-a(b^4+c^4)],](/images/equations/ApolloniusCircle/NumberedEquation5.svg) |
(5)
|
which is Kimberling center 
. Its radius is
 |
(6)
|
where 
is the inradius and 
is the semiperimeter of
the reference triangle. It can be constructed
as the inversive image of the nine-point circle
with respect to the circle orthogonal to the excircles
of the reference triangle. It is a Tucker circle
(Grinberg and Yiu 2002).
Kimberling centers 
for 
,
2038, 3029, 3030, 3031, 3032, 3033, and 3034 lie on the Apollonius circle. It is
also orthogonal to the Stevanović
circle.
