Second Neuberg Circle

Author

Eric W. Weisstein
January 22, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{Blue,NeubergTriangle[2][t]},
{Red,NeubergCircle[2][t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[NeubergTriangle[2][t]],{"(Subscript[N, A])'","(Subscript[N, B])'","(Subscript[N, C])'"}}],
{
{Center[NeubergCircle[2][t]],"X",Red}
}
],LabelOffset->.05]
}],AspectRatio->Automatic]

Center

Center[NeubergCircle[2]][[1]]//TraditionalForm

IncidentCenters[KimberlingCenters,Center[NeubergCircle[2]]]//Timing

Radius

CircleRadius[NeubergCircle[2]]//TraditionalForm

Circle Function

CircleFunction[NeubergCircle[2]]//TraditionalForm

IncidentCenters[KimberlingCenters,CyclicTrilinears[CircleFunction[NeubergCircle[2]]]]//Timing

Incident Centers

(ctrs=IncidentCenters[KimberlingCenters,NeubergCircle[2]])//Timing

Derivation

v=TrilinearVertexMatrix[NeubergTriangle[2]]

Factor[Together//@TrilinearEquation@@v]

eqn= (-2 (a^8) (b^4) (α^2)-5 (a^8) (b^2) (c^2) (α^2)-6 (a^6) (b^4) (c^2) (α^2)-2 (a^4) (b^6) (c^2) (α^2)-(a^2) (b^8) (c^2) (α^2)-2 (a^8) (c^4) (α^2)-6 (a^6) (b^2) (c^4) (α^2)-5 (a^4) (b^4) (c^4) (α^2)-2 (a^2) (b^6) (c^4) (α^2)-2 (a^4) (b^2) (c^6) (α^2)-2 (a^2) (b^4) (c^6) (α^2)-(a^2) (b^2) (c^8) (α^2)-2 (a^7) (b^5) α β-2 (a^5) (b^7) α β-6 (a^7) (b^3) (c^2) α β-16 (a^5) (b^5) (c^2) α β-6 (a^3) (b^7) (c^2) α β-3 (a^7) b (c^4) α β-22 (a^5) (b^3) (c^4) α β-22 (a^3) (b^5) (c^4) α β-3 a (b^7) (c^4) α β-6 (a^5) b (c^6) α β-15 (a^3) (b^3) (c^6) α β-6 a (b^5) (c^6) α β+a b (c^10) α β-2 (a^4) (b^8) (β^2)-(a^8) (b^2) (c^2) (β^2)-2 (a^6) (b^4) (c^2) (β^2)-6 (a^4) (b^6) (c^2) (β^2)-5 (a^2) (b^8) (c^2) (β^2)-2 (a^6) (b^2) (c^4) (β^2)-5 (a^4) (b^4) (c^4) (β^2)-6 (a^2) (b^6) (c^4) (β^2)-2 (b^8) (c^4) (β^2)-2 (a^4) (b^2) (c^6) (β^2)-2 (a^2) (b^4) (c^6) (β^2)-(a^2) (b^2) (c^8) (β^2)-3 (a^7) (b^4) c α γ-6 (a^5) (b^6) c α γ+a (b^10) c α γ-6 (a^7) (b^2) (c^3) α γ-22 (a^5) (b^4) (c^3) α γ-15 (a^3) (b^6) (c^3) α γ-2 (a^7) (c^5) α γ-16 (a^5) (b^2) (c^5) α γ-22 (a^3) (b^4) (c^5) α γ-6 a (b^6) (c^5) α γ-2 (a^5) (c^7) α γ-6 (a^3) (b^2) (c^7) α γ-3 a (b^4) (c^7) α γ+(a^10) b c β γ-6 (a^6) (b^5) c β γ-3 (a^4) (b^7) c β γ-15 (a^6) (b^3) (c^3) β γ-22 (a^4) (b^5) (c^3) β γ-6 (a^2) (b^7) (c^3) β γ-6 (a^6) b (c^5) β γ-22 (a^4) (b^3) (c^5) β γ-16 (a^2) (b^5) (c^5) β γ-2 (b^7) (c^5) β γ-3 (a^4) b (c^7) β γ-6 (a^2) (b^3) (c^7) β γ-2 (b^5) (c^7) β γ-(a^8) (b^2) (c^2) (γ^2)-2 (a^6) (b^4) (c^2) (γ^2)-2 (a^4) (b^6) (c^2) (γ^2)-(a^2) (b^8) (c^2) (γ^2)-2 (a^6) (b^2) (c^4) (γ^2)-5 (a^4) (b^4) (c^4) (γ^2)-2 (a^2) (b^6) (c^4) (γ^2)-6 (a^4) (b^2) (c^6) (γ^2)-6 (a^2) (b^4) (c^6) (γ^2)-2 (a^4) (c^8) (γ^2)-5 (a^2) (b^2) (c^8) (γ^2)-2 (b^4) (c^8) (γ^2));

circ=Circumcircle[eqn];//Timing

ctr=Symmetrize[Center[circ][[1]]]

ToAngles[%]

InputForm[%]

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

Factor[CircleFunction[eqn][[1]]]

InputForm[%]

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

ToAngles[Sqrt[Factor[Distance[CyclicTrilinears[ctr],TrilinearVertexMatrix[NeubergTriangle[2]][[1]]]^2]]]

Refine[%,TriangleQ[]&&Area>0]

Refine[ToAngles[%],TriangleQ[]&&Area>0]

InputForm[%]