Apollonius Circles
There are two 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
(Ogilvy 1990). - 2. The eight Circles (two of which are nondegenerate) which solve Apollonius' Problem for three Circles.
Given one side of a Triangle and the ratio of the lengths of the other two sides, the Locus of the thirdVertex is the Apollonius circle (of the first type) whose Center is on the extension ofthe 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 Circumcircleat 
. 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 theLemoine Line. The circle of Apollonius is also the locus of a point whose Pedal Triangle isIsosceles such that 
.
Let 
and 
be points on the side line 
of a Triangle 
met by the interior and exteriorAngle Bisectors of Angles 
. The Circle with Diameter 
iscalled the -Apollonian circle. Similarly, construct the 
- and 
-Apollonian circles. The Apollonian circles passthrough the Vertices , , and , and through the two Isodynamic Points 
and 
. The Vertices of the D-Triangle lie on the respective Apollonius circles.
References
Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 40 and 294-299, 1929. Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 14-23, 1990.
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