TwoPencilsApolloniusCircles::usage="The circles of Apollonius are formed by two pencils:The first is the ratio r of the distances from one point P to two fixed points C and D, denoted as follows:​ P{x, y} |
PC
PD
= r ​The second is the geometric locus of the P{x, y} that form an inscribed angle θ with the fixed points C and D, it is denoted as follows:​ P{x, y} | C

P
D = θ .References: Public domain; Personal adaptation. ";
Development:
Let the fixed points C{-5, 0} and D{5, 0}:
In[]:=
​​SwatchLegend[{Blue,Red},{"First pencil","Second pencil"}]​​ShowTable[ContourPlot[
2
(x+5)
+
2
y
==
2
r
(
2
(x-5)
+
2
y
),{x,-10,10},{y,-10,10},ContourStyle{Blue,Thick},MaxRecursion3],{r,Union[-((Range[9]/2-5)/(Range[9]/2+5)),{1},-((Range[9]/2+5)/(Range[9]/2-5))]}],TableContourPlot[Sin[θ](
2
x
+
2
y
-25)==10yCos[θ],{x,-10,10},{y,-10,10},ContourStyle{Red,Thick}],θ,0,π,
π
12
,Graphics[{Black,PointSize[0.01],Point[{-5,0}]}],Graphics[{Black,PointSize[0.01],Point[{5,0}]}],AxesTrue,ImageSize700,PlotRangeAll,AxesOrigin{0,0},AspectRatio->Automatic
Out[]=
First pencil
Second pencil
Out[]=