Radical Center

Author

Eric W. Weisstein
December 18, 2004

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

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

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

Graphic

circs={Circle[{0,.1},1],Circle[{3,0},.6],Circle[{1,3},1.2]};
Show[Graphics[{
Transpose[{{Red,Blue,Green},circs,Flatten@RadicalLines[circs,5]}],
PointLabels[Join[
{{RadicalCenter[circs],"R"}},
Transpose[{CircleCenter/@circs,{"(Subscript[O, 1])","(Subscript[O, 2])","(Subscript[O, 3])"},{Red,Blue,Green}}]
]]
}],
AspectRatio->Automatic,PlotRange->{{-1,3.7},{-1,4.3}}
]

Triples of Three Circles

All

t=Triangle[Table[Random[Real,{0,1},20],{3},{2}]];
rules=Join[Thread[{a,b,c}->SideLengths[Evaluate[t]]],Thread[{A,B,C}->Angles[Evaluate[t]]]];
kc=KimberlingCenters;

Length[trips=Subsets[CentralCircles,{3}]]

Length[k=Rest[#]/First[#]&/@((Trilinears/@KimberlingCenters)/.TriangleRules/.rules)]//Timing

search::zero="Bad zero encountered `1`";
l={};
Do[
Print[l0={
i,
trips[[i]],
rc=RadicalCenter[trips[[i]]]/.TriangleRules/.rules;
(
pt=Switch[rc,
_?(#=={0,0,0}&),{0,0,0},
Or@@Thread[#==0]&,Message[search:zero,#];rc,
_,Rest[rc]/First[rc]
];
If[pt=={0,0,0},{0},
If[#=={},{},kc[[#]]]&[Flatten[Position[k,_?(#==pt&)]]]
]
)//Timing
}];
AppendTo[l,l0],
{i,Length[trips]}
]

List

Length[x=Select[l,#[[-1,-1]]≠{}&]]

TableForm[{#[[2]],#[[-1,-1,1]]}&/@x,TableDepth->2]

u=Union[#[[-1,-1,1]]&/@x]

CenterName[0]:="0"

{#[[1,1]],CenterName[#[[1,1]]],Rest/@#}&/@Split[Sort[Reverse[{#[[2]],#[[-1,-1,1]]}]&/@x],First[#1]==First[#2]&]

Special Cases

(Subscript[X, 2457]): Bevan circle, Fuhrmann circle, orthocentric circle

Symmetrize[Together//@DistanceTrilinears[ExactTrilinears[RadicalCenter[{BevanCircle,FuhrmannCircle,OrthocentroidalCircle}]][[1]]]]

Symmetrize[CenterFunction[X[2457]]]

(Subscript[X, 2501]): circumcircle, nine-point circle, Taylor circle

CenterName[X[2501]]

((32(a b c)^2)/(Times@@CyclicTrilinears[a^2+b^2-c^2]))Simplify[Factor[DistanceTrilinears[RadicalCenter[{Circumcircle,NinePointCircle,TaylorCircle}]]]Circumradius[]^2,TriangleQ[]]//Simplify

{#,DistanceTrilinears[#]}&[CenterFunction[X[2501]]]

Symmetrize/@LawOfCosines[Symmetrize/@Factor/@(Factor[Together//@DistanceTrilinears[RadicalCenter[{Circumcircle,NinePointCircle,TaylorCircle}]]])]

Trilinears[X[2501]]

IncidentCenters[{X[2501]},RadicalCenter[{Circumcircle,NinePointCircle,TaylorCircle}]]//Timing