The term "Apollonius circle" has several related uses in mathematics:
1. The set of all points whose distances from two distinct fixed points
are in a constant ratio 
, where 
and 
(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 of the first type passing through a polygon 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).
For 
,
the locus in the first definition is the perpendicular
bisector of the two fixed points.
Given one side of a triangle and a ratio different from 1 for 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 scalene 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 line 
with the tangent to the circumcircle
at 
.
The centers 
, 
, and 
are collinear on the polar
of the symmedian point 
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 their respective vertices 
, 
, and 
, and through the two isodynamic
points 
and 
(Kimberling 1998, p. 68). The 
-Apollonius circle has center with
trilinear coordinates
 |
(1)
|
and radius
 |
(2)
|
where 
,
,
,
and 
.
The other two radii follow by cyclic
permutation. When 
, the corresponding locus is the
perpendicular bisector of 
.
Because the Apollonius circles intersect pairwise in the isodynamic points, they share a common radical line
 |
(3)
|
which is the Brocard axis, the central line 
corresponding to Kimberling center 
, the isogonal conjugate
of the Kiepert parabola focus 
. This line passes through the
circumcenter 
and symmedian point 
.
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.
