%%% This file is part of PlanetMath snapshot of 2009-01-12
%%% Primary Title: Apollonius' circle
%%% Primary Category Code: 51-00
%%% Filename: ApolloniusCircle.tex
%%% Version: 3
%%% Owner: drini
%%% Author(s): pahio, drini
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 \textbf{Apollonius' circle}.
The \htmladdnormallink{locus}{http://planetmath.org/encyclopedia/Locus.html} of a \htmladdnormallink{point}{http://planetmath.org/encyclopedia/Point.html} moving so that the \htmladdnormallink{ratio}{http://planetmath.org/encyclopedia/Reduction2.html} of its \htmladdnormallink{distances}{http://planetmath.org/encyclopedia/MetricTopology.html} from two \htmladdnormallink{fixed points}{http://planetmath.org/encyclopedia/FixedPoint3.html} is \htmladdnormallink{fixed}{http://planetmath.org/encyclopedia/Fixed3.html}, is a \htmladdnormallink{circle}{http://planetmath.org/encyclopedia/Center8.html}.
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If two circles $C_1$ and $C_2$ are fixed with \htmladdnormallink{radii}{http://planetmath.org/encyclopedia/Center8.html} $r_1$ and $r_2$, then the circle of Apollonius of the two \htmladdnormallink{centers}{http://planetmath.org/encyclopedia/Center8.html} with ratio $r_1/r_2$ is the circle whose \htmladdnormallink{diameter}{http://planetmath.org/encyclopedia/Diameter2.html} is the \htmladdnormallink{segment}{http://planetmath.org/encyclopedia/Segment.html} that joins the two \htmladdnormallink{homothety centers}{http://planetmath.org/encyclopedia/HomothetyCenter.html} of the circles.

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