McCay Circles Radical Circle

Author

Eric W. Weisstein
December 21, 2004

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

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

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

Figure

t=Triangle[{{0.3003486278546172`,0.9968270191534088`},{0.34871628293357826`,0.3112766532600932`},{0.4092962066585701`,0.8637410206740449`}}];
Show[Graphics[{
t,
Circumcircle[t],
{Blue,McCayCircles[t]},
{Red,McCayCirclesRadicalCircle[t],
RadicalLines[McCayCircles[t]]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,{Blue,LabelOffset->{-.08,-.02}}]&/@Transpose[{Center/@McCayCircles[t],{"(Subscript[O, A])","(Subscript[O, B])","(Subscript[O, C])"}}],
{{Center[McCayCirclesRadicalCircle[t]],"X",{Red,LabelOffset->{.02,-.08}}}}
],LabelOffset->.06]
}],AspectRatio->Automatic,PlotRange->All]

Orthogonality

Factor[Together//@ToSidelengths[OrthogonalQ[Circle[McCayCirclesRadicalCircle],#]&/@Circles[McCayCircles]]]

Imaginary When Any Two Circles Intersect

ts=Table[RandomTriangle[],{10}]

({#,{a,b,c},McCayCirclesRadicalCircle[#]}/.Thread[{a,b,c}->SideLengths[Evaluate[#]]])&/@ts//ColumnForm

Circle Function

CircleFunction[McCayCirclesRadicalCircle]

IncidentCenters[KimberlingCenters,CircleFunction[McCayCirclesRadicalCircle]]//Timing

Center

Center[McCayCirclesRadicalCircle]

IncidentCenters[KimberlingCenters,Center[McCayCirclesRadicalCircle]]//Timing

Incident Centers

IncidentCenters[KimberlingCenters,McCayCirclesRadicalCircle]//Timing

Radius

CircleRadius[McCayCirclesRadicalCircle]

Derivation

eqns=Numerator[Factor[Refine[Factor[Together//@ToSidelengths[TrilinearEquation/@Circles[McCayCircles]]],TriangleQ[]]]]

lmn=Factor[CircleFunction/@eqns]

ctr=Factor[Symmetrize/@Factor[RadicalCenter[lmn]]]

ctr={-((5 (a^6)-6 (a^4) (b^2)+2 (b^6)-6 (a^4) (c^2)-18 (a^2) (b^2) (c^2)-3 (b^4) (c^2)-3 (b^2) (c^4)+2 (c^6))/a),-((2 (a^6)-6 (a^2) (b^4)+5 (b^6)-3 (a^4) (c^2)-18 (a^2) (b^2) (c^2)-6 (b^4) (c^2)-3 (a^2) (c^4)+2 (c^6))/b),-((2 (a^6)-3 (a^4) (b^2)-3 (a^2) (b^4)+2 (b^6)-18 (a^2) (b^2) (c^2)-6 (a^2) (c^4)-6 (b^2) (c^4)+5 (c^6))/c)}

InputForm[ctr[[1]]]

IncidentCenters[KimberlingCenters,ctr]//Timing

circs=Circles[McCayCircles]

Factor[Refine[Map[Factor,Refine[Sqrt[Together//@ToSidelengths[Distance[ctr,Center[#]]^2-CircleRadius[#]^2&[circs[[1]]]]],TriangleQ[]],{2}],TriangleQ[]]]

InputForm[%]

Factor[Simplify[Together//@ToSidelengths[TrilinearEquation[Circle[McCayCirclesRadicalCircle]]],TriangleQ[]]]

Symmetrize[%]

Factor[CircleFunction[%10]]