Euler-Gergonne-Soddy Circle

Author

Eric W. Weisstein
January 21, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/Euler-Gergonne-SoddyCircle.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Euler-Gergonne-SoddyCircle.html.

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

Figure

t=Triangle[{{0,0},{1,0},{1,2}/10}]//N[#,20]&;
Show[Graphics[{
t,
{Blue,EulerGergonneSoddyTriangle[t],
PerpSquare[Line/@Partition[Vertices[EulerGergonneSoddyTriangle[t]][[{3,1,2}]],2,1],.04,4],
EqualSideTicksAboutMidpoint[Line[Coordinates/@{EvansPoint[t],DeLongchampsPoint[t]}],1,.02]
},
{Red,EulerGergonneSoddyCircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[EulerGergonneSoddyTriangle[t]],{"Fl","Ev","L"}}],
{
{Center[EulerGergonneSoddyCircle[t]],"X",Red}
}
],LabelOffset->.05]
}],AspectRatio->Automatic,PlotRange->All]

Center

Center[EulerGergonneSoddyCircle][[1]]//TraditionalForm

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

Radius

CircleRadius[EulerGergonneSoddyCircle]//TraditionalForm

Circle Function

CircleFunction[EulerGergonneSoddyCircle]//TraditionalForm

IncidentCenters[KimberlingCenters,CyclicTrilinears[CircleFunction[EulerGergonneSoddyCircle]]]//Timing

Incident Centers

(ctrs=IncidentCenters[KimberlingCenters,EulerGergonneSoddyCircle])//Timing

First/@ctrs

Length[ctrs]

{#,CenterName[#]}&/@ctrs//TableForm

Derivation

TrilinearVertexMatrix[EulerGergonneSoddyTriangle]

v=ToSidelengths[HalfAngleFormulas[TrilinearVertexMatrix[EulerGergonneSoddyTriangle]]]

Factor[Together//@TrilinearEquation@@v]

eqn= (2 (a^9) (α^2)-2 (a^8) b (α^2)-(a^7) (b^2) (α^2)-(a^6) (b^3) (α^2)+4 (a^4) (b^5) (α^2)-(a^3) (b^6) (α^2)-(a^2) (b^7) (α^2)-2 (a^8) c (α^2)+3 (a^6) (b^2) c (α^2)-(a^2) (b^6) c (α^2)-(a^7) (c^2) (α^2)+3 (a^6) b (c^2) (α^2)-4 (a^4) (b^3) (c^2) (α^2)+(a^3) (b^4) (c^2) (α^2)+(a^2) (b^5) (c^2) (α^2)-(a^6) (c^3) (α^2)-4 (a^4) (b^2) (c^3) (α^2)+(a^2) (b^4) (c^3) (α^2)+(a^3) (b^2) (c^4) (α^2)+(a^2) (b^3) (c^4) (α^2)+4 (a^4) (c^5) (α^2)+(a^2) (b^2) (c^5) (α^2)-(a^3) (c^6) (α^2)-(a^2) b (c^6) (α^2)-(a^2) (c^7) (α^2)+(a^8) b α β-3 (a^7) (b^2) α β+3 (a^6) (b^3) α β-(a^5) (b^4) α β-(a^4) (b^5) α β+3 (a^3) (b^6) α β-3 (a^2) (b^7) α β+a (b^8) α β-3 (a^7) b c α β+3 (a^5) (b^3) c α β+3 (a^3) (b^5) c α β-3 a (b^7) c α β-8 (a^6) b (c^2) α β+8 (a^5) (b^2) (c^2) α β+8 (a^2) (b^5) (c^2) α β-8 a (b^6) (c^2) α β+4 (a^5) b (c^3) α β-16 (a^3) (b^3) (c^3) α β+4 a (b^5) (c^3) α β+5 (a^4) b (c^4) α β-7 (a^3) (b^2) (c^4) α β-7 (a^2) (b^3) (c^4) α β+5 a (b^4) (c^4) α β+9 (a^3) b (c^5) α β+9 a (b^3) (c^5) α β+2 (a^2) b (c^6) α β+2 a (b^2) (c^6) α β-10 a b (c^7) α β-(a^7) (b^2) (β^2)-(a^6) (b^3) (β^2)+4 (a^5) (b^4) (β^2)-(a^3) (b^6) (β^2)-(a^2) (b^7) (β^2)-2 a (b^8) (β^2)+2 (b^9) (β^2)-(a^6) (b^2) c (β^2)+3 (a^2) (b^6) c (β^2)-2 (b^8) c (β^2)+(a^5) (b^2) (c^2) (β^2)+(a^4) (b^3) (c^2) (β^2)-4 (a^3) (b^4) (c^2) (β^2)+3 a (b^6) (c^2) (β^2)-(b^7) (c^2) (β^2)+(a^4) (b^2) (c^3) (β^2)-4 (a^2) (b^4) (c^3) (β^2)-(b^6) (c^3) (β^2)+(a^3) (b^2) (c^4) (β^2)+(a^2) (b^3) (c^4) (β^2)+(a^2) (b^2) (c^5) (β^2)+4 (b^4) (c^5) (β^2)-a (b^2) (c^6) (β^2)-(b^3) (c^6) (β^2)-(b^2) (c^7) (β^2)+(a^8) c α γ-3 (a^7) b c α γ-8 (a^6) (b^2) c α γ+4 (a^5) (b^3) c α γ+5 (a^4) (b^4) c α γ+9 (a^3) (b^5) c α γ+2 (a^2) (b^6) c α γ-10 a (b^7) c α γ-3 (a^7) (c^2) α γ+8 (a^5) (b^2) (c^2) α γ-7 (a^3) (b^4) (c^2) α γ+2 a (b^6) (c^2) α γ+3 (a^6) (c^3) α γ+3 (a^5) b (c^3) α γ-16 (a^3) (b^3) (c^3) α γ-7 (a^2) (b^4) (c^3) α γ+9 a (b^5) (c^3) α γ-(a^5) (c^4) α γ+5 a (b^4) (c^4) α γ-(a^4) (c^5) α γ+3 (a^3) b (c^5) α γ+8 (a^2) (b^2) (c^5) α γ+4 a (b^3) (c^5) α γ+3 (a^3) (c^6) α γ-8 a (b^2) (c^6) α γ-3 (a^2) (c^7) α γ-3 a b (c^7) α γ+a (c^8) α γ-10 (a^7) b c β γ+2 (a^6) (b^2) c β γ+9 (a^5) (b^3) c β γ+5 (a^4) (b^4) c β γ+4 (a^3) (b^5) c β γ-8 (a^2) (b^6) c β γ-3 a (b^7) c β γ+(b^8) c β γ+2 (a^6) b (c^2) β γ-7 (a^4) (b^3) (c^2) β γ+8 (a^2) (b^5) (c^2) β γ-3 (b^7) (c^2) β γ+9 (a^5) b (c^3) β γ-7 (a^4) (b^2) (c^3) β γ-16 (a^3) (b^3) (c^3) β γ+3 a (b^5) (c^3) β γ+3 (b^6) (c^3) β γ+5 (a^4) b (c^4) β γ-(b^5) (c^4) β γ+4 (a^3) b (c^5) β γ+8 (a^2) (b^2) (c^5) β γ+3 a (b^3) (c^5) β γ-(b^4) (c^5) β γ-8 (a^2) b (c^6) β γ+3 (b^3) (c^6) β γ-3 a b (c^7) β γ-3 (b^2) (c^7) β γ+b (c^8) β γ-(a^7) (c^2) (γ^2)-(a^6) b (c^2) (γ^2)+(a^5) (b^2) (c^2) (γ^2)+(a^4) (b^3) (c^2) (γ^2)+(a^3) (b^4) (c^2) (γ^2)+(a^2) (b^5) (c^2) (γ^2)-a (b^6) (c^2) (γ^2)-(b^7) (c^2) (γ^2)-(a^6) (c^3) (γ^2)+(a^4) (b^2) (c^3) (γ^2)+(a^2) (b^4) (c^3) (γ^2)-(b^6) (c^3) (γ^2)+4 (a^5) (c^4) (γ^2)-4 (a^3) (b^2) (c^4) (γ^2)-4 (a^2) (b^3) (c^4) (γ^2)+4 (b^5) (c^4) (γ^2)-(a^3) (c^6) (γ^2)+3 (a^2) b (c^6) (γ^2)+3 a (b^2) (c^6) (γ^2)-(b^3) (c^6) (γ^2)-(a^2) (c^7) (γ^2)-(b^2) (c^7) (γ^2)-2 a (c^8) (γ^2)-2 b (c^8) (γ^2)+2 (c^9) (γ^2));

circ=Circumcircle[eqn];//Timing

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

t=Triangle[{{0,0},{1,0},{1,2}/10}]//N[#,20]&;

Cartesian[Center[circ][t]]

Cartesian[Midpoint[EvansPoint,DeLongchampsPoint][t]]

Symmetrize[ToSidelengths[Midpoint[EvansPoint,DeLongchampsPoint]][[1]]]

Symmetrize[Center[circ][[1]]]

InputForm[%]

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

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

InputForm[%]

Simplify[Factor[CircleRadius[circ]^2],TriangleQ[]]

...ran overnight...

Factor[ToSidelengths[HalfAngleFormulas[TrilinearVertexMatrix[EulerGergonneSoddyTriangle]]]]

ToAngles[Factor[Distance[CyclicTrilinears[ctr],Factor[ToSidelengths[HalfAngleFormulas[TrilinearVertexMatrix[EulerGergonneSoddyTriangle]]]][[1]]]^2]]

Refine[Sqrt[ToAngles[-((((5 (a^5)-3 (a^4) b-2 (a^3) (b^2)-2 (a^2) (b^3)-3 a (b^4)+5 (b^5)-3 (a^4) c+6 (a^2) (b^2) c-3 (b^4) c-2 (a^3) (c^2)+6 (a^2) b (c^2)+6 a (b^2) (c^2)-2 (b^3) (c^2)-2 (a^2) (c^3)-2 (b^2) (c^3)-3 a (c^4)-3 b (c^4)+5 (c^5)))^2) ((a^6)-(a^4) (b^2)-(a^2) (b^4)+(b^6)-(a^4) (c^2)+3 (a^2) (b^2) (c^2)-(b^4) (c^2)-(a^2) (c^4)-(b^2) (c^4)+(c^6)))/(16 (a-b-c) (a+b-c) (a-b+c) (a+b+c) (((2 (a^5)-(a^4) b-(a^3) (b^2)-(a^2) (b^3)-a (b^4)+2 (b^5)-(a^4) c+2 (a^2) (b^2) c-(b^4) c-(a^3) (c^2)+2 (a^2) b (c^2)+2 a (b^2) (c^2)-(b^3) (c^2)-(a^2) (c^3)-(b^2) (c^3)-a (c^4)-b (c^4)+2 (c^5)))^2))]],TriangleQ[]&&Area>0]

InputForm[%]

Factor[ToSidelengths[Distance[Midpoint[EvansPoint,DeLongchampsPoint],Trilinears[DeLongchampsPoint]]^2]]