Steiner Circumellipse

Author

Eric W. Weisstein
January 30, 2005

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

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

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

Equation

Circumconic parameters

{x,y,z}=1/{a,b,c};

Trilinear Equations

TrilinearEquation[SteinerCircumellipse]

Center

{x(-a x+b y+c z),y(a x-b y+c z),z(a x+b y-c z)}

CenterFunction[Centroid]

Derivation as Conjugates

TrilinearEquation[LineAtInfinity]/.Thread[{α,β,γ}->IsotomicConjugate[{α,β,γ}]]

((#)/(a b c))&/@TrilinearEquation[LemoineAxis]/.Thread[{α,β,γ}->IsogonalConjugate[{α,β,γ}]]

Graphic

Solve[1/(a α)+1/(b β)+1/(c γ)==0/.γ->1,β]

t=Triangle[{{0,0},{1,0},{.1,.7}}];
Show[Graphics[{
t,
{Red,SteinerCircumellipse[t]},
{Blue,Circumcircle[t]},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
{
{Centroid[t],"G",Red},
{SteinerPoint[t],"S",Purple},
{Circumcenter[t],"O",Blue}
}
],
{PointSize[.025],LabelOffset->.04}
]
}],
PlotRange->All,AspectRatio->Automatic]

Axes

Conway's Construction

t=Triangle[{{0,0},{1,0},{.1,.7}}];
is=Intersections[ParryCircle[t],LemoineAxis[t]];
lines=Line[Coordinates/@{Centroid[t],#}]&/@is;
Show[Graphics[{
t,
{Red,SteinerCircumellipse[t]},
{Blue,Line[t1=Coordinates/@Through[{SymmedianPoint,Centroid,SteinerPoint}[t]]],
EqualAngleDivisionTicks[t1,2,1,.1,.02,.2],
AngleBisector[t1]
},
{Green,Line[t2=Coordinates/@Through[{Circumcenter,Centroid,TarryPoint}[t]]],
EqualAngleDivisionTicks[t2,2,2,.1,.02,.2],
AngleBisector[t2]
},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
{
{Centroid[t],"G",{Red,LabelOffset->{.06,-.03}}},
{SymmedianPoint[t],"K",{Blue,LabelOffset->{-.06,0}}},
{SteinerPoint[t],"S",Blue},
{TarryPoint[t],"T",Green},
{Circumcenter[t],"O",Green}
}
],
{PointSize[.02],LabelOffset->.03}
]
}],PlotRange->{{-.3,1.2},{-.3,.9}},
AspectRatio->Automatic]

Moses' Construction

t=Triangle[{{0,0},{1,0},{.1,.7}}];
is=Intersections[ParryCircle[t],LemoineAxis[t]];
lines=Line[Coordinates/@{Centroid[t],#}]&/@is;
Show[Graphics[{
t,
{Red,
SteinerCircumellipse[t],
lines,
PerpSquare[lines,.1,3]
},
{Blue,ParryCircle[t]},
{Green,LemoineAxis[t]},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{is,{"X","X'"}}],
{
{Centroid[t],"G",Red},
{X[351][t],"(Subscript[X, 351])",Green}
}
],
{PointSize[.02],LabelOffset->.1}
]
}],PlotRange->{{-2,2},{-1.5,1}},
AspectRatio->Automatic]

Axes

Explicit Solution

eqn1=Numerator[Factor[TrilinearEquation[ParryCircle]]]

TrilinearEquation[LemoineAxis]

solns1={α,β,γ}/.FullSimplify[Solve[{eqn1==0,TrilinearEquation[LemoineAxis]==0,(a α+b  β+c  γ)==2Area//ToSidelengths},{α,β,γ}],TriangleQ[]]

solns={{(a ((Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) ((2 (a^2)-(b^2)-(c^2))+√ ((a^4)+(b^4)-(b^2) (c^2)+(c^4)-(a^2) ((b^2)+(c^2))))))/(6 (a-b) (a+b) (a-c) (a+c)),(b ((Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) (-((a^2)-2 (b^2)+(c^2))+√ ((a^4)+(b^4)-(b^2) (c^2)+(c^4)-(a^2) ((b^2)+(c^2))))))/(6 (-(a^2)+(b^2)) (b-c) (b+c)),(c ((Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) (((a^2)+(b^2)-2 (c^2))-√((a^4)+(b^4)-(b^2) (c^2)+(c^4)-(a^2) ((b^2)+(c^2))))))/(6 (a-c) (a+c) (-(b^2)+(c^2)))},{-(a ((Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) ((-2 (a^2)+(b^2)+(c^2))+√((a^4)+(b^4)-(b^2) (c^2)+(c^4)-(a^2) ((b^2)+(c^2))))))/(6 (a-b) (a+b) (a-c) (a+c)),-(b ((Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) (((a^2)-2 (b^2)+(c^2))+√ ((a^4)+(b^4)-(b^2) (c^2)+(c^4)-(a^2) ((b^2)+(c^2))))))/(6 (-(a^2)+(b^2)) (b-c) (b+c)),(c ((Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) (((a^2)+(b^2)-2 (c^2))+√((a^4)+(b^4)-(b^2) (c^2)+(c^4)-(a^2) ((b^2)+(c^2))))))/(6 (a-c) (a+c) (-(b^2)+(c^2)))}};

Simplify[solns1-solns,TriangleQ[]]

Greater@@@Table[Simplify[Distance[Trilinears[Centroid],#]&/@solns/.Thread[{a,b,c}->SideLengths[Evaluate[RandomTriangle[]]]],Area>0],{10}]

Together//@ToSidelengths[PerpendicularQ@@(TrilinearLine[Line[Trilinears[Centroid],#]]&/@solns)]

eqn2=Numerator[Together[TrilinearEquation[SteinerCircumellipse]]]

lines=Symmetrize[Factor[TrilinearLine[Line[#,Trilinears[Centroid]]][[1]].{α,β,γ}]]&/@solns

sqrt=(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4);
lines2=Symmetrize[(coef=Coefficient[#,(Sqrt[sqrt])])^2sqrt-Expand[#-coef(Sqrt[sqrt])]^2]&/@lines

Equal@@lines2

eqn3=lines2[[1]];

soln2=Together//@({α,β,γ}/.Solve[{
eqn2==0,
eqn3==0,
a α+b β+c γ==2Area//ToSidelengths
},{α,β,γ}])

Dimensions[soln2]

Together//@((TrilinearEquation[SteinerCircumellipse]/.Thread[{α,β,γ}->#])&/@soln2)

Refine[Distance[#,Trilinears[Centroid]]&/@soln2/.{a->1,b->1.2,c->1.4},Area>0]

ab=Together//@ToSidelengths[Distance[Trilinears[Centroid],#]^2&/@soln2[[{1,3}]]]

Algebraic Equations

eqn1=Numerator[Factor[TrilinearEquation[ParryCircle]]]

eqn2=TrilinearEquation[LemoineAxis]

16ToSidelengths[Area]^2

Gröbner Basis

Factor/@GroebnerBasis[
{
eqn1,
eqn2,
a α+b β+c γ-2area,
16area^2-(a+b-c) (a-b+c) (-a+b+c) (a+b+c)
},
{α,β,γ},area,MonomialOrder->EliminationOrder]

Resultants

gamma=Solve[a α+b β+c γ==2Area,γ]

eqn11=Numerator[Factor[Numerator[Together[eqn1/.gamma[[1]]]]/.Area^2:>ToSidelengths[Area^2]]]

eqn21=Numerator[Together[eqn2/.gamma[[1]]]]

Factor[Resultant[eqn11,eqn21,β]/.Area^2:>ToSidelengths[Area^2]]

alpha=α/.FullSimplify[Solve[((a^6)-2 (a^4) (b^2)+(a^2) (b^4)-2 (a^4) (c^2)-2 (a^2) (b^2) (c^2)+(a^2) (c^4)+32 (a^3) Area α-16 a Area (b^2) α-16 a Area (c^2) α-12 (a^4) (α^2)+12 (a^2) (b^2) (α^2)+12 (a^2) (c^2) (α^2)-12 (b^2) (c^2) (α^2))==0,α],TriangleQ[]&&Area>0]

alphas=Factor[Refine[Factor//@ToAngles[alpha],Area>0]]

betas=Factor[Flatten[β/.(Solve[(eqn2/.gamma[[1]])==0,β]/.α:>#)&/@alphas]]

Z=(Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]);
z1={
a(2a^2-b^2-c^2-Z)(b^2-c^2),
b (-a^2+2 b^2-c^2-Z)(c^2-a^2),
c (-a^2-b^2+2c^2-Z)(a^2-b^2)};
z2={
a(2a^2-b^2-c^2+Z)(b^2-c^2),
b (-a^2+2 b^2-c^2+Z)(c^2-a^2),
c (-a^2-b^2+2c^2+Z)(a^2-b^2)};
t=Triangle[{{0,0},{1,0},{.1,.7}}];
Show[Graphics[{
t,
{Red,SteinerCircumellipse[t]},
{Green,LemoineAxis[t]},
{Blue,ParryCircle[t]},
{Orange,Line[Coordinates/@{Cartesian[z1[t]],Centroid[t],Cartesian[z2[t]]}]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{
{Centroid[t],"G",Red},
{Cartesian[z1[t]],"(Subscript[Z, 1])",Red},
{Cartesian[z2[t]],"(Subscript[Z, 2])",Red}
}
],PointSize[.02]]
}],AspectRatio->Automatic,PlotRange->{{-2,2},{-1.7,1.3}}]

eqn1=Numerator[Together[TrilinearEquation[Line[z1,Trilinears[Centroid]]]]]

eqn11=coef=Coefficient[eqn1,(Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])]

eqn12=Expand[eqn1-coef (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])]

Factor[eqn11^2((a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4))-eqn12^2]

eqn10= ((a^2) (b^2) (α^2)-(a^2) (c^2) (α^2)+2 (a^3) b α β-2 a (b^3) α β-(a^2) (b^2) (β^2)+(b^2) (c^2) (β^2)-2 (a^3) c α γ+2 a (c^3) α γ+2 (b^3) c β γ-2 b (c^3) β γ+(a^2) (c^2) (γ^2)-(b^2) (c^2) (γ^2));

eqn2=Numerator[Together[TrilinearEquation[SteinerCircumellipse]]]

gb=Factor/@GroebnerBasis[{eqn10,eqn2,(a α+b β+c γ)^2-ToSidelengths[(2Area)^2]},{α},{β,γ},MonomialOrder->EliminationOrder][[1]]

terms=Factor[Coefficient[gb,Sequence@@#]&/@({α,2#}&/@Range[0,4])]

Factor[terms.α^Range[0,8,2]-gb]

alphaeqn=terms.α^Range[0,8,2]

gb=Factor/@GroebnerBasis[{eqn10,eqn2,(a α+b β+c γ)^2-ToSidelengths[(2Area)^2]},{β},{α,γ},MonomialOrder->EliminationOrder][[1]]

terms=Factor[Coefficient[gb,Sequence@@#]&/@({β,2#}&/@Range[0,4])]

Factor[terms.β^Range[0,8,2]-gb]

betaeqn=terms.β^Range[0,8,2]

gb=Factor/@GroebnerBasis[{eqn10,eqn2,(a α+b β+c γ)^2-ToSidelengths[(2Area)^2]},{γ},{α,β},MonomialOrder->EliminationOrder][[1]]

terms=Factor[Coefficient[gb,Sequence@@#]&/@({γ,2#}&/@Range[0,4])]

Factor[terms.γ^Range[0,8,2]-gb]

gammaeqn=terms.γ^Range[0,8,2]

NRoots[(gb==0)/.Thread[{a,b,c}->SideLengths[Evaluate[t]]],α]

soln={α,β,γ}/.NSolve[#==0&/@{eqn10,eqn2,(a α+b β+c γ)^2-ToSidelengths[(2Area)^2]}/.Thread[{a,b,c}->SideLengths[Evaluate[t]]],{α,β,γ}]

t=Triangle[{{0,0},{1,0},{.1,.7}}];
Show[Graphics[{
t,
{Red,SteinerCircumellipse[t]},
{Green,LemoineAxis[t]},
{Blue,ParryCircle[t]},
{Orange,Line[Coordinates/@{Cartesian[z1[t]],Centroid[t],Cartesian[z2[t]]}]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{
{Cartesian[soln[[1]][t]],"(Subscript[X, 1])",Blue},
{Cartesian[soln[[3]][t]],"(Subscript[X, 2])",Blue},
{Cartesian[soln[[5]][t]],"(Subscript[X, 3])",Blue},
{Cartesian[soln[[7]][t]],"(Subscript[X, 4])",Blue},
{Centroid[t],"G",Red},
{Cartesian[z1[t]],"(Subscript[Z, 1])",Red},
{Cartesian[z2[t]],"(Subscript[Z, 2])",Red}
}
],PointSize[.02]]
}],AspectRatio->Automatic,PlotRange->{{-2,2},{-1.7,1.3}}]

{alphaeqn,betaeqn,gammaeqn}/.((a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)):>𝒵^2

soln={α,β,γ}/.Solve[#==0&/@{eqn10,eqn2,(a α+b β+c γ)-2Area},{α,β,γ}];//Timing

Dimensions[soln]

t=Triangle[{{0,0},{1,0},{1,7}/10}]//N[#,20]&;

pts=Cartesian[#[t]]&/@ToSidelengths[soln/.𝒵:>(Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])]

expr1=Together//@(Together//@(soln[[1]]/.Area^n_:>Area^Mod[n,2]ToSidelengths[Area]^(n-Mod[n,2]))/.Area^n_:>Area^Mod[n,2]ToSidelengths[Area]^(n-Mod[n,2]))

Cartesian[ToSidelengths[expr1/.𝒵:>(Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])][t]]

expr2=Factor//@expr1

Cartesian[ToSidelengths[expr2/.𝒵:>(Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])][t]]

expr3=Refine[expr2,TriangleQ[]]/.(poly=√(((a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)) (-2 (a^8) c+6 (a^6) (b^2) c-8 (a^4) (b^4) c+6 (a^2) (b^6) c-2 (b^8) c+6 (a^6) (c^3)-2 (a^4) (b^2) (c^3)-2 (a^2) (b^4) (c^3)+6 (b^6) (c^3)-8 (a^4) (c^5)-2 (a^2) (b^2) (c^5)-8 (b^4) (c^5)+6 (a^2) (c^7)+6 (b^2) (c^7)-2 (c^9)+(a+b-c) c (a-b+c) (-a+b+c) (a+b+c) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]) Abs[(a^2)+(b^2)-2 (c^2)]))):>p

expr4=Factor[expr3]

Factor//@ToAngles[FullSimplify[poly^2,TriangleQ[]]]

Refine[Sqrt[Factor[Refine[%84/.(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4):>(𝒵^2)/.Expand[2((a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4))]:>2(𝒵^2),𝒵>0]]],Area>0&&𝒵>0]

expr5=Factor[Factor[expr4/.(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4):>(𝒵^2)/.p:>Refine[Sqrt[Factor[Refine[%84/.(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4):>(𝒵^2)/.Expand[2((a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4))]:>2(𝒵^2),𝒵>0]]],Area>0&&𝒵>0]]/.(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4):>(𝒵^2)]

Cartesian[ToSidelengths[expr5/.𝒵:>(Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])][t]]

expr6=Simplify[expr5,TriangleQ[]&&Area>0&&𝒵>0]

Cartesian[ToSidelengths[expr6/.𝒵:>(Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])][t]]

expr7=Factor[{(2 Area (-Abs[(a^2)+(b^2)-2 (c^2)] (Sqrt[(𝒵^5) (2 𝒵+Abs[(a^2)+(b^2)-2 (c^2)])])+((a^2)+(b^2)-2 (c^2)) ((a^6)-(b^6)+(b^4) (c^2)-(b^2) (c^4)-(a^4) (2 (b^2)+(c^2))-(c^2) (Sqrt[(𝒵^3) (2 𝒵+Abs[(a^2)+(b^2)-2 (c^2)])])+(a^2) (2 (b^4)+(c^4)+(Sqrt[(𝒵^3) (2 𝒵+Abs[(a^2)+(b^2)-2 (c^2)])])))))/(3 a ((a^2)-(b^2)) ((a^2)+(b^2)-2 (c^2)) (𝒵^2)),(2 Area (-Abs[(a^2)+(b^2)-2 (c^2)] (Sqrt[(𝒵^5) (2 𝒵+Abs[(a^2)+(b^2)-2 (c^2)])])-((a^2)+(b^2)-2 (c^2)) ((a^6)-(b^6)+(b^4) (c^2)-(a^4) (2 (b^2)+(c^2))+(a^2) (2 (b^4)+(c^4))+(c^2) (Sqrt[(𝒵^3) (2 𝒵+Abs[(a^2)+(b^2)-2 (c^2)])])-(b^2) ((c^4)+(Sqrt[(𝒵^3) (2 𝒵+Abs[(a^2)+(b^2)-2 (c^2)])])))))/(3 b (-(a^2)+(b^2)) ((a^2)+(b^2)-2 (c^2)) (𝒵^2)),-((2 Area (-1+(Sqrt[2+((Abs[(a^2)+(b^2)-2 (c^2)])/𝒵)])))/(3 c))}/.Abs[x_]:>-x]

expr8=Refine[expr7,TriangleQ[]&&𝒵>0]

Cartesian[ToSidelengths[expr8/.𝒵:>(Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])][t]]

expr9={(1/( a (a-b) (a+b) (𝒵^2))) ((a^6)-2 (a^4) (b^2)+2 (a^2) (b^4)-(b^6)-(a^4) (c^2)+(b^4) (c^2)+(a^2) (c^4)-(b^2) (c^4)+𝒵(Sqrt[-(a^2) 𝒵-(b^2) 𝒵+2 (c^2) 𝒵+2 (𝒵^2)])((a^2) -(c^2) +𝒵)),-(1/((a-b) b (a+b) (𝒵^2))) (-(a^6)+2 (a^4) (b^2)-2 (a^2) (b^4)+(b^6)+(a^4) (c^2)-(b^4) (c^2)-(a^2) (c^4)+(b^2) (c^4)+𝒵(Sqrt[-(a^2) 𝒵-(b^2) 𝒵+2 (c^2) 𝒵+2 (𝒵^2)])((b^2)-(c^2) +𝒵)),-(( (-1+((Sqrt[-(a^2)-(b^2)+2 (c^2)+2 𝒵])/(Sqrt[𝒵]))))/( c))}

Cartesian[ToSidelengths[expr9/.𝒵:>(Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])][t]]

expr9={(Factor[(a^6)-2 (a^4) (b^2)+2 (a^2) (b^4)-(b^6)-(a^4) (c^2)+(b^4) (c^2)+(a^2) (c^4)-(b^2) (c^4)]+𝒵 ((a^2)-(c^2)+𝒵) (Sqrt[-(a^2) 𝒵-(b^2) 𝒵+2 (c^2) 𝒵+2 (𝒵^2)]))/(a (a-b) (a+b) (𝒵^2)),(Factor[-(-(a^6)+2 (a^4) (b^2)-2 (a^2) (b^4)+(b^6)+(a^4) (c^2)-(b^4) (c^2)-(a^2) (c^4)+(b^2) (c^4))]-𝒵 ((b^2)-(c^2)+𝒵) (Sqrt[-(a^2) 𝒵-(b^2) 𝒵+2 (c^2) 𝒵+2 (𝒵^2)]))/((a-b) b (a+b) (𝒵^2)),-((-1+((Sqrt[-(a^2)-(b^2)+2 (c^2)+2 𝒵])/(Sqrt[𝒵])))/c)}/.(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4):>(𝒵^2)

Cartesian[ToSidelengths[expr9/.𝒵:>(Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])][t]]

Simplify[expr9]

expr10={(1/a)(1+((((a^2)-(c^2)+𝒵) (Sqrt[𝒵 (-(a^2)-(b^2)+2 ((c^2)+𝒵))]))/( (a-b) (a+b) 𝒵))),(1/b)(1+((-((b^2)-(c^2)+𝒵) (Sqrt[𝒵 (-(a^2)-(b^2)+2 ((c^2)+𝒵))]))/((a-b)  (a+b) 𝒵))),((1-((Sqrt[-(a^2)-(b^2)+2 ((c^2)+𝒵)])/(Sqrt[𝒵])))/c)}

Cartesian[ToSidelengths[expr10/.𝒵:>(Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])][t]]

Factor[Factor[Expand[(((((((a^2)-(c^2)+𝒵) (Sqrt[𝒵 (-(a^2)-(b^2)+2 ((c^2)+𝒵))]))/((a-b) (a+b) 𝒵))))^2)]/.(𝒵^2):>(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]/.(𝒵^2):>(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]

Factor[Factor[Expand[((((((-(b^2)+(c^2)-𝒵) (Sqrt[𝒵 (-(a^2)-(b^2)+2 ((c^2)+𝒵))]))/((a-b) (a+b) 𝒵))))^2)]/.(𝒵^2):>(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]/.(𝒵^2):>(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]

expr11={((1+(Sqrt[((2 (a^2)-(b^2)-(c^2)+2 𝒵)/𝒵)]))/a),((1-(Sqrt[-(((a^2)-2 (b^2)+(c^2)-2 𝒵)/𝒵)]))/b),((1-((Sqrt[-(a^2)-(b^2)+2 ((c^2)+𝒵)])/(Sqrt[𝒵])))/c)};

Cartesian[ToSidelengths[expr11/.𝒵:>(Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])][t]]

expr11={(1/a)(1+(Sqrt[((2((a^2)+𝒵)-((b^2)+(c^2)))/𝒵)])),(1/b)(1-(Sqrt[((2((b^2)+𝒵)-((c^2)+(a^2)))/𝒵)])),(1/c)(1-(Sqrt[((2 ((c^2)+𝒵)-((a^2)+(b^2)))/𝒵)]))};

Cartesian[ToSidelengths[expr11/.𝒵:>(Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])][t]]

pts=Cartesian[#[t]]&/@ToSidelengths[soln/.𝒵:>(Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])]

Cartesian[ToSidelengths[{(1/a)(1-(Sqrt[((2((a^2)+𝒵)-((b^2)+(c^2)))/𝒵)])),(1/b)(1+(Sqrt[((2((b^2)+𝒵)-((c^2)+(a^2)))/𝒵)])),(1/c)(1+(Sqrt[((2 ((c^2)+𝒵)-((a^2)+(b^2)))/𝒵)]))}/.𝒵:>(Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])][t]]

v={{(1/a)(1+(Sqrt[((2((a^2)+𝒵)-((b^2)+(c^2)))/𝒵)])),(1/b)(1-(Sqrt[((2((b^2)+𝒵)-((c^2)+(a^2)))/𝒵)])),(1/c)(1-(Sqrt[((2 ((c^2)+𝒵)-((a^2)+(b^2)))/𝒵)]))},
{(1/a)(1-(Sqrt[((2((a^2)+𝒵)-((b^2)+(c^2)))/𝒵)])),(1/b)(1+(Sqrt[((2((b^2)+𝒵)-((c^2)+(a^2)))/𝒵)])),(1/c)(1+(Sqrt[((2 ((c^2)+𝒵)-((a^2)+(b^2)))/𝒵)]))}};

vpts=Cartesian[#[t]]&/@(v/.𝒵:>(Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))

expr7b=Factor[{(2 Area (-Abs[(a^2)+(b^2)-2 (c^2)] (Sqrt[(𝒵^5) (2 𝒵+Abs[(a^2)+(b^2)-2 (c^2)])])+((a^2)+(b^2)-2 (c^2)) ((a^6)-(b^6)+(b^4) (c^2)-(b^2) (c^4)-(a^4) (2 (b^2)+(c^2))-(c^2) (Sqrt[(𝒵^3) (2 𝒵+Abs[(a^2)+(b^2)-2 (c^2)])])+(a^2) (2 (b^4)+(c^4)+(Sqrt[(𝒵^3) (2 𝒵+Abs[(a^2)+(b^2)-2 (c^2)])])))))/(3 a ((a^2)-(b^2)) ((a^2)+(b^2)-2 (c^2)) (𝒵^2)),(2 Area (-Abs[(a^2)+(b^2)-2 (c^2)] (Sqrt[(𝒵^5) (2 𝒵+Abs[(a^2)+(b^2)-2 (c^2)])])-((a^2)+(b^2)-2 (c^2)) ((a^6)-(b^6)+(b^4) (c^2)-(a^4) (2 (b^2)+(c^2))+(a^2) (2 (b^4)+(c^4))+(c^2) (Sqrt[(𝒵^3) (2 𝒵+Abs[(a^2)+(b^2)-2 (c^2)])])-(b^2) ((c^4)+(Sqrt[(𝒵^3) (2 𝒵+Abs[(a^2)+(b^2)-2 (c^2)])])))))/(3 b (-(a^2)+(b^2)) ((a^2)+(b^2)-2 (c^2)) (𝒵^2)),-((2 Area (-1+(Sqrt[2+((Abs[(a^2)+(b^2)-2 (c^2)])/𝒵)])))/(3 c))}/.Abs[x_]:>x]

vv={{(1/a)(1-(Sqrt[((2(-(a^2)+𝒵)+((b^2)+(c^2)))/𝒵)])),(1/b)(1+(Sqrt[((2(-(b^2)+𝒵)+((c^2)+(a^2)))/𝒵)])),(1/c)(1-(Sqrt[((2 (-(c^2)+𝒵)+((a^2)+(b^2)))/𝒵)]))},
{(1/a)(1+(Sqrt[((2(-(a^2)+𝒵)+((b^2)+(c^2)))/𝒵)])),(1/b)(1-(Sqrt[((2(-(b^2)+𝒵)+((c^2)+(a^2)))/𝒵)])),(1/c)(1+(Sqrt[((2 (-(c^2)+𝒵)+((a^2)+(b^2)))/𝒵)]))}};

vvpts=Cartesian[#[t]]&/@(vv/.𝒵:>(Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))

pts=Cartesian[#[t]]&/@ToSidelengths[soln/.𝒵:>(Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])]

t=Triangle[{{0,0},{1,0},{.1,.7}}];
Show[Graphics[{
t,
{Red,SteinerCircumellipse[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{
{vvpts[[1]],"(Subscript[X, 1])",Blue},
{vvpts[[2]],"(Subscript[X, 2])",Blue},
{vpts[[1]],"(Subscript[X, 3])",Blue},
{vpts[[2]],"(Subscript[X, 4])",Blue},
{Centroid[t],"G",Red}
}
],PointSize[.02],LabelOffset->.04]
}],AspectRatio->Automatic]

expr=Together//@ToSidelengths[OrthogonalQ[TrilinearLine[Line@@v],TrilinearLine[Line@@vv]]]/.Sqrt[x_]Sqrt[y_]:>Simplify[Sqrt[x y],TriangleQ[]]

Simplify[Refine[expr,𝒵>0],a>0&&b>0&&c>0&&𝒵>0]

t=Triangle[{{0,0},{1,0},{1,7}/10}]//N[#,20]&;

expr/.𝒵:>(Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

{TraditionalForm/@vv,TraditionalForm/@v}

Area

area=Factor[Simplify[π (Sqrt[ Times@@ab])]]

Simplify[area(Δ/Area)//ToSidelengths,TriangleQ[]]

Incident Centers

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

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

Check

Short/@(FullSimplify[ToSidelengths[TrigExpand[TrilinearEquation[SteinerCircumellipse]/.Thread[{α,β,γ}->Trilinears[#]]]],
TriangleQ[]]&/@ctrs)//Timing

Circumhyperbola

CenterName[X[#]]&/@Range[5]

Factor[ToSidelengths[Conic[{Sequence@@IdentityMatrix[3],Sequence@@(Trilinears/@{Orthocenter,X[1]})}]]]

eqn1= ((a^2) α β-(b^2) α β-a c α β+b c α β-(a^2) α γ+a b α γ-b c α γ+(c^2) α γ-a b β γ+(b^2) β γ+a c β γ-(c^2) β γ);

Factor[TrilinearEquation[SteinerCircumellipse]]

eqn2=a b α β+a c α γ+b c β γ;

Solve[{
eqn1==0,
eqn2==0,
a α+b β+c γ==2Area//ToSidelengths
},{α,β,γ}]

int=Factor[(1/(2a b c α β γ))IsogonalConjugate[ToSidelengths[ Intersections[TrilinearLine[Line[Trilinears[Circumcenter],IsogonalConjugate[{α,β,γ}]]],TrilinearLine[LemoineAxis]]]]]

Factor[2 a b c ToAngles[1/((a^2) (Factor[((a^2) b-(b^3)-b (c^2))] α β +Factor[((a^2) c -(b^2) c -(c^3))]α γ+2 a b c β γ))]]

Symmetrize/@(int/.Thread[{α,β,γ}->Trilinears[Incenter]])

IncidentCenters[KimberlingCenters,int/.Thread[{α,β,γ}->Trilinears[X[1]]]]//Timing

IncidentCenters[KimberlingCenters,int/.Thread[{α,β,γ}->Trilinears[X[2]]]]//Timing

IncidentCenters[KimberlingCenters,int/.Thread[{α,β,γ}->Trilinears[X[3]]]]//Timing

IncidentCenters[KimberlingCenters,int/.Thread[{α,β,γ}->Trilinears[X[5]]]]//Timing

NoName, P = X(5), -- on lines {5,99},{53,648},{311,670},{316,1263}.

CenterName/@X/@{2471,671,290}//ColumnForm