Neuberg Circles Radical Circle

Author

Eric W. Weisstein
December 9, 2004

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

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

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

Center

ctr=Symmetrize/@RadicalCenter[lmn]

IncidentCenters[KimberlingCenters,CircleCenter[NeubergCirclesRadicalCircle[1]]]//Timing

CenterName[X[194]]

Radius

circ=Circles[NeubergCircles[1]][[1]]

Factor/@Simplify[Sqrt[Factor[Together//@(Distance[ctr,CircleCenter[circ=Circles[NeubergCircles[1]][[1]]]]^2-CircleRadius[circ]^2)]],TriangleQ[]]

InputForm[%27]

Orthogonality Check

Together//@OrthogonalQ[rc=Circle[{(((a^2) (b^2)+(a^2) (c^2)-(b^2) (c^2))/a),(((a^2) (b^2)-(a^2) (c^2)+(b^2) (c^2))/b),((-(a^2) (b^2)+(a^2) (c^2)+(b^2) (c^2))/c)},(1/((a^2) (b^2)+(a^2) (c^2)+(b^2) (c^2)))(√((a^6) (b^4)+(a^4) (b^6)-2 (a^6) (b^2) (c^2)+(a^4) (b^4) (c^2)-2 (a^2) (b^6) (c^2)+(a^6) (c^4)+(a^4) (b^2) (c^4)+(a^2) (b^4) (c^4)+(b^6) (c^4)+(a^4) (c^6)-2 (a^2) (b^2) (c^6)+(b^4) (c^6)))],#]&/@Circles[NeubergCircles[1]]

Circle Functon

Factor[Simplify[Factior[Together//@ToSidelengths[CircleEquation[rc]]],TriangleQ[]]]

Factor[CircleFunction[%]]

IncidentCenters[KimberlingCenters,CyclicTrilinears[CircleFunction[NeubergCirclesRadicalCircle[1]]]]//Timing

Figure

t=Triangle[{{0,0},{1,0},{.8,.6}}];
Show[Graphics[{
t,
{Red,NeubergCircles[1][t]},
{Blue,NeubergCirclesRadicalCircle[1][t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{CircleCenter/@NeubergCircles[1][t],{"(Subscript[N, A])","(Subscript[N, B])","(Subscript[N, C])"}}],
{{CircleCenter[NeubergCirclesRadicalCircle[1][t]],"(Subscript[X, 194])",Blue}}
],LabelOffset->.06,PointSize[.025]]
}],AspectRatio->Automatic]

Incident Centers

IncidentCenters[KimberlingCenters,NeubergCirclesRadicalCircle[1]]//Timing