Apollonius Circle
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)],](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/ApolloniusCircle/NumberedEquation5.gif) |
(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.
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