Apollonius Circle

Author

Eric W. Weisstein
December 23, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/ApolloniusCircle.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/ApolloniusCircle.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Second Type

c1=Circle@@@{{{-1,0},.8},{{1,1},.3},{{.9,-.6},.5}};
ac=ApolloniusCircles[c1]//Flatten;
Show[GraphicsArray[Partition[Table[Show[Graphics[{ac[[i]],Disk@@@c1}],		
	DisplayFunction->Identity,AspectRatio->Automatic,
PlotRange->{{-2,2.5},{-2,4}}],{i,8}],4]],
	DisplayFunction->$DisplayFunction]

Show[Graphics[{Disk@@@c1,ac}],AspectRatio->Automatic]

Geometric Construction

c1=Circle@@@{{{-1,0},.8},{{1,1},.3},{{.9,-.6},.5}};
aip=ApolloniusInversePoints[c1];
ac=ApolloniusIntersectionPoints[c1];
Show[Graphics[{
HomotheticLines[c1,1.5],
c1,
Table[Text[ToString[i],Coordinates[aip[[j,i]]]-{.1,.1}],{i,4},{j,3}],
{
PointSize[.015],
{
PointSize[.03],
RadicalCenter[c1]
},
HomotheticPoints[c1],
Table[{GrayLevel[.33(i-1)],aip[[i]]},{i,3}],
{
RGBColor[1,0,0],
Flatten[ac,2]
}
}
} 
],PlotRange->All,AspectRatio->Automatic]

Third Type

t=Triangle[{{0,0},{1,0},{.1,.7}}];
Show[Graphics[{
t,
Transpose[{{Red,Green,Blue},ApolloniusCircles[t]}],
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"},{Red,Green,Blue}}],
{
{IsodynamicPoint[1][t],"S"},
{IsodynamicPoint[2][t],"S'"}
}
],PointSize[.018]]
},AspectRatio->Automatic]
]

Centers

Center/@Circles[ApolloniusCircles]

Radius

CircleRadius/@Circles[ApolloniusCircles]//TraditionalForm

Radical Lines are Degenerate

eqns=Factor[Simplify[Together//@ToSidelengths[TrilinearEquation/@Circles[ApolloniusCircles]],TriangleQ[]]]

lmn=Factor[CircleFunction/@eqns]

Factor[RadicalCenter[lmn]]

Factor[Simplify[ToSidelengths[RadicalLine@@Take[Circles[ApolloniusCircles],2]]/.Sqrt[x_]:>FullSimplify[Sqrt[Expand[x]],TriangleQ[]],TriangleQ[]]]

lmn=Factor[((-(a+c)(b+c)(a-c)(b-c))/(a b c))%53[[1]]]

Together[Expand[{-(((-b+c) (b+c))/a),-(((a-c) (a+c))/b),(((a-b) (a+b))/c)}]]

IncidentCenters[KimberlingCenters,lmn]//Timing

CenterName[X[523]]

CenterName[X[110]]

RadicalLines[ApolloniusCircles[t]]

t=Triangle[{{0,0},{1,0},{.7,.5}}];
TrilinearLine[Trilinears[X[523]]][t]

t=Triangle[{{0,0},{1,0},{.3,.5}}];
Show[Graphics[{
t,
{Blue,ApolloniusCircles[t]},
{Red,TrilinearLine[Trilinears[X[523]]][t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{Center/@ApolloniusCircles[t],{"(Subscript[A, A])","(Subscript[A, B])","(Subscript[A, C])"}}],
Append[#,Red]&/@Transpose[{IsodynamicPoint[#][t]&/@{1,2},{"S","S'"}}]
],LabelOffset->.12]
}],AspectRatio->Automatic,PlotRange->{{-2.3,1.5},{-1.3,2.2}}]

Derivation

eqn=Factor[ToSidelengths[Together//@(TrilinearEquation@@Append[Trilinears/@IsodynamicPoint/@{1,2},{1,0,0}]/.Sin[x_]:>TrigExpand[Sin[x]])]/.Sqrt[x_]:>FullSimplify[Sqrt[Factor[x]],TriangleQ[]]/.x_^(3/2):>FullSimplify[Factor[x]^(3/2),TriangleQ[]]]

eqn=((a^2) c α β+(b^2) c α β-(c^3) α β+a b c (β^2)-(a^2) b α γ+(b^3) α γ-b (c^2) α γ-a b c (γ^2))

Factor[((4Center[eqn])/((a+b+c)(b-c)(b+c) (a+b-c)(-a+b-c) (-a+b+c)))]

Refine[Map[Factor,Factor[Together//@ExpandAll[Together//@CircleRadius[eqn]]],{3}] ((Circumradius[])/R),TriangleQ[]]

InputForm[%%]

Fourth Type

Figure

t=Triangle[{{0,0},{1,0},{1/4,.7}}];
n=FullSimplify[NinePointCircle[t]];
j=FullSimplify[Excircles[t]];
v=Point/@Vertices[t];
apt=ApolloniusPoint[t];
Show[Graphics[{
{Dashing[{.02}],Sides[t,1]},
t,
{Blue,j=Excircles[t]},
{Red,ap=ApolloniusOuterCircle[j]},
abc=First/@(Intersections[ap,#]&/@j),
PointLabels[Join[
Transpose[{v,{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{abc,{"A'","B'","C'"}}],
],PointSize[.022]
]
},AspectRatio->Automatic],
ImageSize->250
]

Circle Function

Factor[FullSimplify[Together//@DistanceTrilinears[CircleFunction[CircleEquation[CyclicTrilinears[CenterFunction[X[970]]],(r^2+s^2)/(4r)/.r->Inradius[]/.s->Semiperimeter[]]]],TriangleQ[]]]

IncidentCenters[KimberlingCenters,CyclicTrilinears[CircleFunction[ApolloniusCircle]]]

CenterFunction[X[940]]

Center

CenterFunction[X[970]]

FullSimplify[Together//@DistanceTrilinears[CircleCenter[ApolloniusCircle][[1]]]]

IncidentCenters[KimberlingCenters,CircleCenter[ApolloniusCircle]]

Radius

N[(Inradius[]^2+Semiperimeter[]^2)/(4Inradius[])/.Thread[{a,b,c}->SideLengths[Evaluate[t]]],20]

N[DistanceTrilinears[CircleRadius[ApolloniusCircle]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]],20]

LeafCount[ra2=Simplify[(Together//@(DistanceTrilinears[CircleRadius[ApolloniusCircle]^2]/.(Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])->Δ))/.Δ^2:>(a+b-c) (a-b+c) (-a+b+c) (a+b+c),TriangleQ[]]]

LeafCount[ra=FullSimplify[Sqrt[ra2],TriangleQ[]]]

ra

Together//@(((Inradius[]^2+Semiperimeter[]^2)/(4Inradius[]))^2-ra^2)

Incident Centers

IncidentCenters[KimberlingCenters,ApolloniusCircle]//Timing

{#,CenterName[#],CenterFunction[#]}&/@X/@{2037,2038,3029,3030,3031,3032,3033,3034}//TableForm

TraditionalForm[alpha=Sqrt[1-4 Sin[BrocardAngle]^2]a Cot[A]-(2CircleRadius[ApolloniusCircle]-Sqrt[1-4 Sin[BrocardAngle]^2]Semiperimeter Csc[BrocardAngle])Cos[A+BrocardAngle]]

IncidentCenters[{X[2037]},CyclicTrilinears[alpha]//ToSidelengths]

t=Triangle[{{0,0},{1,0},{.3,5}}];
{X[2037][t],Cartesian[CyclicTrilinears[alpha][t]]}

Intersections[ApolloniusCircle[t],BrocardAxis[t]]

Orthogonality

Check

Together//@DistanceTrilinears[First[OrthogonalQ[ApolloniusCircle,StevanovicCircle]]]//Timing

Graphic

t=Triangle[{{0,0},{1,0},{.2,.7}}];
circs={ApolloniusCircle,StevanovicCircle};
centers={"(Subscript[X, 970])","(Subscript[X, 650])"};
colors={Red,Green};
Show[Graphics[{
Transpose[{colors,Through[circs[t]]}],
PointLabels[Join[
Transpose[{CircleCenter[#[t]]&/@circs,centers,colors}]
],LabelOffset->.2
]
}],AspectRatio->Automatic]