First Neuberg Circle

Author

Eric W. Weisstein
January 22, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{Blue,NeubergTriangle[1][t]},
{Red,NeubergCircle[1][t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[NeubergTriangle[1][t]],{"(Subscript[N, A])","(Subscript[N, B])","(Subscript[N, C])"}}],
{
{Center[NeubergCircle[1][t]],"X",Red}
}
],LabelOffset->.05]
}],AspectRatio->Automatic]

Center

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

IncidentCenters[KimberlingCenters,Center[NeubergCircle[1]]]//Timing

Radius

CircleRadius[NeubergCircle[1]]//TraditionalForm

Circle Function

CircleFunction[NeubergCircle[1]]//TraditionalForm

TeXForm[%35]

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

Incident Centers

(ctrs=IncidentCenters[KimberlingCenters,NeubergCircle[1]])//Timing

Derivation

v=TrilinearVertexMatrix[NeubergTriangle[1]]

Factor[Together//@TrilinearEquation@@v]

eqn=((a^8) (b^4) (α^2)+(a^4) (b^8) (α^2)+(a^8) (b^2) (c^2) (α^2)-(a^6) (b^4) (c^2) (α^2)+(a^8) (c^4) (α^2)-(a^6) (b^2) (c^4) (α^2)-(a^4) (b^4) (c^4) (α^2)-(a^2) (b^6) (c^4) (α^2)-(a^2) (b^4) (c^6) (α^2)+(a^4) (c^8) (α^2)+(a^9) (b^3) α β+(a^7) (b^5) α β+(a^5) (b^7) α β+(a^3) (b^9) α β-(a^7) b (c^4) α β+(a^5) (b^3) (c^4) α β+(a^3) (b^5) (c^4) α β-a (b^7) (c^4) α β+(a^5) b (c^6) α β+3 (a^3) (b^3) (c^6) α β+a (b^5) (c^6) α β+(a^8) (b^4) (β^2)+(a^4) (b^8) (β^2)-(a^4) (b^6) (c^2) (β^2)+(a^2) (b^8) (c^2) (β^2)-(a^6) (b^2) (c^4) (β^2)-(a^4) (b^4) (c^4) (β^2)-(a^2) (b^6) (c^4) (β^2)+(b^8) (c^4) (β^2)-(a^4) (b^2) (c^6) (β^2)+(b^4) (c^8) (β^2)-(a^7) (b^4) c α γ+(a^5) (b^6) c α γ+(a^9) (c^3) α γ+(a^5) (b^4) (c^3) α γ+3 (a^3) (b^6) (c^3) α γ+(a^7) (c^5) α γ+(a^3) (b^4) (c^5) α γ+a (b^6) (c^5) α γ+(a^5) (c^7) α γ-a (b^4) (c^7) α γ+(a^3) (c^9) α γ+(a^6) (b^5) c β γ-(a^4) (b^7) c β γ+3 (a^6) (b^3) (c^3) β γ+(a^4) (b^5) (c^3) β γ+(b^9) (c^3) β γ+(a^6) b (c^5) β γ+(a^4) (b^3) (c^5) β γ+(b^7) (c^5) β γ-(a^4) b (c^7) β γ+(b^5) (c^7) β γ+(b^3) (c^9) β γ-(a^6) (b^4) (c^2) (γ^2)-(a^4) (b^6) (c^2) (γ^2)+(a^8) (c^4) (γ^2)-(a^4) (b^4) (c^4) (γ^2)+(b^8) (c^4) (γ^2)-(a^4) (b^2) (c^6) (γ^2)-(a^2) (b^4) (c^6) (γ^2)+(a^4) (c^8) (γ^2)+(a^2) (b^2) (c^8) (γ^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

FullSimplify[Refine[Factor[CircleRadius[circ]^2],TriangleQ[]],TriangleQ[]]

...ran overnight...

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

ToAngles[%]

InputForm[%]