Bickart Points

Author

Eric W. Weisstein
April 1, 2006

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

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

©2006 Wolfram Research, Inc. except for portions noted otherwise

Distances

QQ= Sqrt[2 (a^2 b^2 c^2 Z^3) -2 (b^4 c^4 (2 S^2 - a^2 SW) +c^4 a^4 (2 S^2 - b^2 SW) +a^4 b^4 (2 S^2 - c^2 SW))];
Z=Sqrt[a^4+b^4+c^4-b^2 c^2-c^2 a^2-a^2 b^2];

cc=With[{a=(1/3)Sqrt[(a^2)+(b^2)+(c^2)+2Z],b=(1/3)Sqrt[(a^2)+(b^2)+(c^2)-2Z]},(Sqrt[(a^2)-(b^2)])]//Simplify//Factor

ToAngles[Factor[ToSidelengths[2 (b^4 c^4 (2 S^2 - a^2 SW) +c^4 a^4 (2 S^2 - b^2 SW) +a^4 b^4 (2 S^2 - c^2 SW))]]]

Obtuse

t=RandomTriangle[MachinePrecision,Type->"Obtuse"];
rules=Flatten[{Thread[{a,b,c}->SideLengths[Evaluate[t]]],Thread[{A,B,C}->Angles[Evaluate[t]]]}]

AcuteQ[t]

{cc,(2/3)(Sqrt[Z])}/.rules

ToSidelengths[Distance[CyclicTrilinears[(QQ+2#(b^2 - c^2) (a^4 - b^2 c^2 - a^2 Z))/a],Trilinears[Centroid]]]&/@{-1,1}/.rules

{Cartesian[CyclicTrilinears[(QQ-2(b^2 - c^2) (a^4 - b^2 c^2 - a^2 Z))/a][t]],Cartesian[CyclicTrilinears[(QQ+2(b^2 - c^2) (a^4 - b^2 c^2 - a^2 Z))/a][t]]}

Cartesian[#[t]]&/@(CyclicTrilinears/@Trilinears[BickartPoint[#]]&/@{1,2})

ToSidelengths[
Table[Distance[{
(((Sqrt[2])Area)/a)+(-1)^i(Sqrt[-(a^2)+Z+b c Cos[B-C]]),(((Sqrt[2])Area)/b)+(-1)^j(Sqrt[-(b^2)+Z+a c Cos[A-C]]),(((Sqrt[2])Area)/c)+(-1)^k(Sqrt[-(c^2)+Z+a b Cos[A-B]])},Trilinears[Centroid]],{i,2},{j,2},{k,2}]]/.rules

Obtuse

t=RandomTriangle[MachinePrecision,Type->"Obtuse"];
rules=Flatten[{Thread[{a,b,c}->SideLengths[Evaluate[t]]],Thread[{A,B,C}->Angles[Evaluate[t]]]}]

AcuteQ[t]

{cc,(2/3)(Sqrt[Z])}/.rules

ToSidelengths[Distance[CyclicTrilinears[(QQ+2#(b^2 - c^2) (a^4 - b^2 c^2 - a^2 Z))/a],Trilinears[Centroid]]]&/@{-1,1}/.rules

ToSidelengths[
Table[Distance[{
(((Sqrt[2])Area)/a)+(-1)^i(Sqrt[-(a^2)+Z+b c Cos[B-C]]),
(((Sqrt[2])Area)/b)+(-1)^j(Sqrt[-(b^2)+Z+a c Cos[A-C]]),
(((Sqrt[2])Area)/c)+(-1)^k(Sqrt[-(c^2)+Z+a b Cos[A-B]])},Trilinears[Centroid]],{i,2},{j,2},{k,2}]]/.rules

Figure

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]]];

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}];

{x1,x2,x3,x4}=Cartesian[#[t]]&/@soln[[{1,3,5,7}]];

t=Triangle[{{0,0},{1,0},{.1,.7}}];
Show[Graphics[{
t,
{Red,SteinerCircumellipse[t]},
{Blue,Line[Coordinates/@{x1,x2}]},
{Green,Line[Coordinates/@{x3,x4}]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Transpose[{Cartesian[Trilinears[BickartPoint[#]][t]]&/@{1,2},{"(Subscript[F, 1])","(Subscript[F, 2])"},{Blue,Blue}}],
{
{Centroid[t],"G",Red},
{x1,"(Subscript[X, 1])",Red},
{x2,"(Subscript[X, 2])",Red},
{x3,"(Subscript[X, 3])",Red},
{x4,"(Subscript[X, 4])",Red}
}
],PointSize[.02],LabelOffset->.04]
}],AspectRatio->Automatic]

Derivation

soln2={{(1/(6 a ((a^2)-(b^2))))((a^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])-(b^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])+(a^2) c √(((a^6)-(a^4) (b^2)-(a^2) (b^4)+(b^6)-4 (a^4) (c^2)-4 (b^4) (c^2)+5 (a^2) (c^4)+5 (b^2) (c^4)-2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))-(c^3) √(((a^6)-(a^4) (b^2)-(a^2) (b^4)+(b^6)-4 (a^4) (c^2)-4 (b^4) (c^2)+5 (a^2) (c^4)+5 (b^2) (c^4)-2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))+c (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]) √(((a^6)-(a^4) (b^2)-(a^2) (b^4)+(b^6)-4 (a^4) (c^2)-4 (b^4) (c^2)+5 (a^2) (c^4)+5 (b^2) (c^4)-2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))),(1/(6 b (-(a^2)+(b^2))))(-(a^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])+(b^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])+(b^2) c √(((a^6)-(a^4) (b^2)-(a^2) (b^4)+(b^6)-4 (a^4) (c^2)-4 (b^4) (c^2)+5 (a^2) (c^4)+5 (b^2) (c^4)-2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))-(c^3) √(((a^6)-(a^4) (b^2)-(a^2) (b^4)+(b^6)-4 (a^4) (c^2)-4 (b^4) (c^2)+5 (a^2) (c^4)+5 (b^2) (c^4)-2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))+c (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]) √(((a^6)-(a^4) (b^2)-(a^2) (b^4)+(b^6)-4 (a^4) (c^2)-4 (b^4) (c^2)+5 (a^2) (c^4)+5 (b^2) (c^4)-2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))),(1/(6 c))((Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])-c √(((a^6)-(a^4) (b^2)-(a^2) (b^4)+(b^6)-4 (a^4) (c^2)-4 (b^4) (c^2)+5 (a^2) (c^4)+5 (b^2) (c^4)-2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))))},{(1/(6 a ((a^2)-(b^2))))((a^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])-(b^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])-(a^2) c √(((a^6)-(a^4) (b^2)-(a^2) (b^4)+(b^6)-4 (a^4) (c^2)-4 (b^4) (c^2)+5 (a^2) (c^4)+5 (b^2) (c^4)-2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))+(c^3) √(((a^6)-(a^4) (b^2)-(a^2) (b^4)+(b^6)-4 (a^4) (c^2)-4 (b^4) (c^2)+5 (a^2) (c^4)+5 (b^2) (c^4)-2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))-c (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]) √(((a^6)-(a^4) (b^2)-(a^2) (b^4)+(b^6)-4 (a^4) (c^2)-4 (b^4) (c^2)+5 (a^2) (c^4)+5 (b^2) (c^4)-2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))),(1/(6 b (-(a^2)+(b^2))))(-(a^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])+(b^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])-(b^2) c √(((a^6)-(a^4) (b^2)-(a^2) (b^4)+(b^6)-4 (a^4) (c^2)-4 (b^4) (c^2)+5 (a^2) (c^4)+5 (b^2) (c^4)-2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))+(c^3) √(((a^6)-(a^4) (b^2)-(a^2) (b^4)+(b^6)-4 (a^4) (c^2)-4 (b^4) (c^2)+5 (a^2) (c^4)+5 (b^2) (c^4)-2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))-c (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]) √(((a^6)-(a^4) (b^2)-(a^2) (b^4)+(b^6)-4 (a^4) (c^2)-4 (b^4) (c^2)+5 (a^2) (c^4)+5 (b^2) (c^4)-2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))),(1/(6 c))((Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])+c √(((a^6)-(a^4) (b^2)-(a^2) (b^4)+(b^6)-4 (a^4) (c^2)-4 (b^4) (c^2)+5 (a^2) (c^4)+5 (b^2) (c^4)-2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))))},{(1/(6 a ((a^2)-(b^2))))((a^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])-(b^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])+(a^2) c √((-(a^6)+(a^4) (b^2)+(a^2) (b^4)-(b^6)+4 (a^4) (c^2)+4 (b^4) (c^2)-5 (a^2) (c^4)-5 (b^2) (c^4)+2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))-(c^3) √((-(a^6)+(a^4) (b^2)+(a^2) (b^4)-(b^6)+4 (a^4) (c^2)+4 (b^4) (c^2)-5 (a^2) (c^4)-5 (b^2) (c^4)+2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))-c (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]) √((-(a^6)+(a^4) (b^2)+(a^2) (b^4)-(b^6)+4 (a^4) (c^2)+4 (b^4) (c^2)-5 (a^2) (c^4)-5 (b^2) (c^4)+2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))),(1/(6 b (-(a^2)+(b^2))))(-(a^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])+(b^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])+(b^2) c √((-(a^6)+(a^4) (b^2)+(a^2) (b^4)-(b^6)+4 (a^4) (c^2)+4 (b^4) (c^2)-5 (a^2) (c^4)-5 (b^2) (c^4)+2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))-(c^3) √((-(a^6)+(a^4) (b^2)+(a^2) (b^4)-(b^6)+4 (a^4) (c^2)+4 (b^4) (c^2)-5 (a^2) (c^4)-5 (b^2) (c^4)+2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))-c (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]) √((-(a^6)+(a^4) (b^2)+(a^2) (b^4)-(b^6)+4 (a^4) (c^2)+4 (b^4) (c^2)-5 (a^2) (c^4)-5 (b^2) (c^4)+2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))),(1/(6 c))((Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])-c √((-(a^6)+(a^4) (b^2)+(a^2) (b^4)-(b^6)+4 (a^4) (c^2)+4 (b^4) (c^2)-5 (a^2) (c^4)-5 (b^2) (c^4)+2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))))},{(1/(6 a ((a^2)-(b^2))))((a^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])-(b^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])-(a^2) c √((-(a^6)+(a^4) (b^2)+(a^2) (b^4)-(b^6)+4 (a^4) (c^2)+4 (b^4) (c^2)-5 (a^2) (c^4)-5 (b^2) (c^4)+2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))+(c^3) √((-(a^6)+(a^4) (b^2)+(a^2) (b^4)-(b^6)+4 (a^4) (c^2)+4 (b^4) (c^2)-5 (a^2) (c^4)-5 (b^2) (c^4)+2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))+c (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]) √((-(a^6)+(a^4) (b^2)+(a^2) (b^4)-(b^6)+4 (a^4) (c^2)+4 (b^4) (c^2)-5 (a^2) (c^4)-5 (b^2) (c^4)+2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))),(1/(6 b (-(a^2)+(b^2))))(-(a^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])+(b^2) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])-(b^2) c √((-(a^6)+(a^4) (b^2)+(a^2) (b^4)-(b^6)+4 (a^4) (c^2)+4 (b^4) (c^2)-5 (a^2) (c^4)-5 (b^2) (c^4)+2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))+(c^3) √((-(a^6)+(a^4) (b^2)+(a^2) (b^4)-(b^6)+4 (a^4) (c^2)+4 (b^4) (c^2)-5 (a^2) (c^4)-5 (b^2) (c^4)+2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))+c (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]) √((-(a^6)+(a^4) (b^2)+(a^2) (b^4)-(b^6)+4 (a^4) (c^2)+4 (b^4) (c^2)-5 (a^2) (c^4)-5 (b^2) (c^4)+2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])))),(1/(6 c))((Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])+c √((-(a^6)+(a^4) (b^2)+(a^2) (b^4)-(b^6)+4 (a^4) (c^2)+4 (b^4) (c^2)-5 (a^2) (c^4)-5 (b^2) (c^4)+2 (c^6)-2 (a^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (b^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (b^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (a^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])+4 (b^2) (c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])-2 (c^4) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))/((c^2) (Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]))))}};

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

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

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

Sqrt[Simplify[-Subtract@@ab/.(Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)]):>𝒵]]

soln={α,β,γ}/.Solve[{
((ExactDistance[{α,β,γ},ExactTrilinears[Centroid]])^2)==(4/9)𝒵,
Det[{
{α,β,γ},
Trilinears[Centroid],
soln2[[3]]/.(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4):>(𝒵^2)
}]==0,
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)])]

Distance[Centroid[t],#]&/@pts

(2/3)(Sqrt[𝒵])/.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]]]

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"}}],
Transpose[{pts,{"(Subscript[F, 1])","(Subscript[F, 2])"},{Red,Red}}],
{
{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}}]

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)])]

expr1=Together//@(soln[[1]]/.{f_Cos:>ToSidelengths[f],(Sqrt[(𝒵^2)]):>𝒵})

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=Together//@(%147/.𝒵^n_:>𝒵^Mod[n,2]((a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4))^((n-Mod[n,2])/2)/.Area^n_:>Area^Mod[n,2]ToSidelengths[Area]^(n-Mod[n,2]))

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=expr2/.(poly=-(a^28) (c^2)+12 (a^26) (b^2) (c^2)-68 (a^24) (b^4) (c^2)+242 (a^22) (b^6) (c^2)-606 (a^20) (b^8) (c^2)+1132 (a^18) (b^10) (c^2)-1629 (a^16) (b^12) (c^2)+1836 (a^14) (b^14) (c^2)-1629 (a^12) (b^16) (c^2)+1132 (a^10) (b^18) (c^2)-606 (a^8) (b^20) (c^2)+242 (a^6) (b^22) (c^2)-68 (a^4) (b^24) (c^2)+12 (a^2) (b^26) (c^2)-(b^28) (c^2)+7 (a^26) (c^4)-65 (a^24) (b^2) (c^4)+275 (a^22) (b^4) (c^4)-693 (a^20) (b^6) (c^4)+1125 (a^18) (b^8) (c^4)-1139 (a^16) (b^10) (c^4)+490 (a^14) (b^12) (c^4)+490 (a^12) (b^14) (c^4)-1139 (a^10) (b^16) (c^4)+1125 (a^8) (b^18) (c^4)-693 (a^6) (b^20) (c^4)+275 (a^4) (b^22) (c^4)-65 (a^2) (b^24) (c^4)+7 (b^26) (c^4)-25 (a^24) (c^6)+188 (a^22) (b^2) (c^6)-646 (a^20) (b^4) (c^6)+1356 (a^18) (b^6) (c^6)-1991 (a^16) (b^8) (c^6)+2296 (a^14) (b^10) (c^6)-2356 (a^12) (b^12) (c^6)+2296 (a^10) (b^14) (c^6)-1991 (a^8) (b^16) (c^6)+1356 (a^6) (b^18) (c^6)-646 (a^4) (b^20) (c^6)+188 (a^2) (b^22) (c^6)-25 (b^24) (c^6)+59 (a^22) (c^8)-361 (a^20) (b^2) (c^8)+989 (a^18) (b^4) (c^8)-1579 (a^16) (b^6) (c^8)+1528 (a^14) (b^8) (c^8)-636 (a^12) (b^10) (c^8)-636 (a^10) (b^12) (c^8)+1528 (a^8) (b^14) (c^8)-1579 (a^6) (b^16) (c^8)+989 (a^4) (b^18) (c^8)-361 (a^2) (b^20) (c^8)+59 (b^22) (c^8)-100 (a^20) (c^10)+498 (a^18) (b^2) (c^10)-1093 (a^16) (b^4) (c^10)+1422 (a^14) (b^6) (c^10)-1319 (a^12) (b^8) (c^10)+1184 (a^10) (b^10) (c^10)-1319 (a^8) (b^12) (c^10)+1422 (a^6) (b^14) (c^10)-1093 (a^4) (b^16) (c^10)+498 (a^2) (b^18) (c^10)-100 (b^20) (c^10)+126 (a^18) (c^12)-514 (a^16) (b^2) (c^12)+881 (a^14) (b^4) (c^12)-819 (a^12) (b^6) (c^12)+326 (a^10) (b^8) (c^12)+326 (a^8) (b^10) (c^12)-819 (a^6) (b^12) (c^12)+881 (a^4) (b^14) (c^12)-514 (a^2) (b^16) (c^12)+126 (b^18) (c^12)-119 (a^16) (c^14)+404 (a^14) (b^2) (c^14)-534 (a^12) (b^4) (c^14)+404 (a^10) (b^6) (c^14)-310 (a^8) (b^8) (c^14)+404 (a^6) (b^10) (c^14)-534 (a^4) (b^12) (c^14)+404 (a^2) (b^14) (c^14)-119 (b^16) (c^14)+83 (a^14) (c^16)-245 (a^12) (b^2) (c^16)+237 (a^10) (b^4) (c^16)-75 (a^8) (b^6) (c^16)-75 (a^6) (b^8) (c^16)+237 (a^4) (b^10) (c^16)-245 (a^2) (b^12) (c^16)+83 (b^14) (c^16)-41 (a^12) (c^18)+114 (a^10) (b^2) (c^18)-87 (a^8) (b^4) (c^18)+28 (a^6) (b^6) (c^18)-87 (a^4) (b^8) (c^18)+114 (a^2) (b^10) (c^18)-41 (b^12) (c^18)+13 (a^10) (c^20)-39 (a^8) (b^2) (c^20)+26 (a^6) (b^4) (c^20)+26 (a^4) (b^6) (c^20)-39 (a^2) (b^8) (c^20)+13 (b^10) (c^20)-2 (a^8) (c^22)+8 (a^6) (b^2) (c^22)-12 (a^4) (b^4) (c^22)+8 (a^2) (b^6) (c^22)-2 (b^8) (c^22)+2 (a^24) (c^4) 𝒵-20 (a^22) (b^2) (c^4) 𝒵+94 (a^20) (b^4) (c^4) 𝒵-276 (a^18) (b^6) (c^4) 𝒵+566 (a^16) (b^8) (c^4) 𝒵-856 (a^14) (b^10) (c^4) 𝒵+980 (a^12) (b^12) (c^4) 𝒵-856 (a^10) (b^14) (c^4) 𝒵+566 (a^8) (b^16) (c^4) 𝒵-276 (a^6) (b^18) (c^4) 𝒵+94 (a^4) (b^20) (c^4) 𝒵-20 (a^2) (b^22) (c^4) 𝒵+2 (b^24) (c^4) 𝒵-12 (a^22) (c^6) 𝒵+88 (a^20) (b^2) (c^6) 𝒵-288 (a^18) (b^4) (c^6) 𝒵+540 (a^16) (b^6) (c^6) 𝒵-592 (a^14) (b^8) (c^6) 𝒵+264 (a^12) (b^10) (c^6) 𝒵+264 (a^10) (b^12) (c^6) 𝒵-592 (a^8) (b^14) (c^6) 𝒵+540 (a^6) (b^16) (c^6) 𝒵-288 (a^4) (b^18) (c^6) 𝒵+88 (a^2) (b^20) (c^6) 𝒵-12 (b^22) (c^6) 𝒵+34 (a^20) (c^8) 𝒵-192 (a^18) (b^2) (c^8) 𝒵+486 (a^16) (b^4) (c^8) 𝒵-744 (a^14) (b^6) (c^8) 𝒵+824 (a^12) (b^8) (c^8) 𝒵-816 (a^10) (b^10) (c^8) 𝒵+824 (a^8) (b^12) (c^8) 𝒵-744 (a^6) (b^14) (c^8) 𝒵+486 (a^4) (b^16) (c^8) 𝒵-192 (a^2) (b^18) (c^8) 𝒵+34 (b^20) (c^8) 𝒵-60 (a^18) (c^10) 𝒵+268 (a^16) (b^2) (c^10) 𝒵-512 (a^14) (b^4) (c^10) 𝒵+528 (a^12) (b^6) (c^10) 𝒵-224 (a^10) (b^8) (c^10) 𝒵-224 (a^8) (b^10) (c^10) 𝒵+528 (a^6) (b^12) (c^10) 𝒵-512 (a^4) (b^14) (c^10) 𝒵+268 (a^2) (b^16) (c^10) 𝒵-60 (b^18) (c^10) 𝒵+72 (a^16) (c^12) 𝒵-260 (a^14) (b^2) (c^12) 𝒵+378 (a^12) (b^4) (c^12) 𝒵-324 (a^10) (b^6) (c^12) 𝒵+268 (a^8) (b^8) (c^12) 𝒵-324 (a^6) (b^10) (c^12) 𝒵+378 (a^4) (b^12) (c^12) 𝒵-260 (a^2) (b^14) (c^12) 𝒵+72 (b^16) (c^12) 𝒵-60 (a^14) (c^14) 𝒵+184 (a^12) (b^2) (c^14) 𝒵-192 (a^10) (b^4) (c^14) 𝒵+68 (a^8) (b^6) (c^14) 𝒵+68 (a^6) (b^8) (c^14) 𝒵-192 (a^4) (b^10) (c^14) 𝒵+184 (a^2) (b^12) (c^14) 𝒵-60 (b^14) (c^14) 𝒵+34 (a^12) (c^16) 𝒵-96 (a^10) (b^2) (c^16) 𝒵+78 (a^8) (b^4) (c^16) 𝒵-32 (a^6) (b^6) (c^16) 𝒵+78 (a^4) (b^8) (c^16) 𝒵-96 (a^2) (b^10) (c^16) 𝒵+34 (b^12) (c^16) 𝒵-12 (a^10) (c^18) 𝒵+36 (a^8) (b^2) (c^18) 𝒵-24 (a^6) (b^4) (c^18) 𝒵-24 (a^4) (b^6) (c^18) 𝒵+36 (a^2) (b^8) (c^18) 𝒵-12 (b^10) (c^18) 𝒵+2 (a^8) (c^20) 𝒵-8 (a^6) (b^2) (c^20) 𝒵+12 (a^4) (b^4) (c^20) 𝒵-8 (a^2) (b^6) (c^20) 𝒵+2 (b^8) (c^20) 𝒵):>p

expr4=Factor[Refine[expr3/.p->Factor[poly],TriangleQ[]]]/.Abs[x_]^2:>x^2

Cartesian[ToSidelengths[expr4/.𝒵:>(Sqrt[(a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4)])][t]]

expr5={
(1/(3 a (a-b) (a+b)))(2Area(a-b) (a+b)+(Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+(a^2) (c^2)+(b^2) (c^2)-2 (c^4)+2 (c^2) 𝒵])((a^2)-(c^2)-𝒵 )),(1/(3b (a-b)  (a+b)))(2Area(a-b)(a+b)+(Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+(a^2) (c^2)+(b^2) (c^2)-2 (c^4)+2 (c^2) 𝒵])((c^2)-(b^2)+𝒵)),((2Area-(Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+(a^2) (c^2)+(b^2) (c^2)-2 (c^4)+2 (c^2) 𝒵]))/(3 c))};

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

Refine[Sqrt[Factor[Expand[(-(a^4)+2 (a^2) (b^2)-(b^4)+(a^2) (c^2)+(b^2) (c^2)-2 (c^4)+2 (c^2) 𝒵)((((a^2)-(c^2)-𝒵 ))^2)]/.𝒵^n_:>𝒵^Mod[n,2]((a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4))^((n-Mod[n,2])/2)]],TriangleQ[]]

Refine[Sqrt[Factor[Expand[(-(a^4)+2 (a^2) (b^2)-(b^4)+(a^2) (c^2)+(b^2) (c^2)-2 (c^4)+2 (c^2) 𝒵)((((c^2)-(b^2)+𝒵))^2)]/.𝒵^n_:>𝒵^Mod[n,2]((a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4))^((n-Mod[n,2])/2)]],TriangleQ[]]

expr5={
(1/(a ))(2Area-(Sqrt[-2 (a^4)+(a^2) (b^2)-(b^4)+(a^2) (c^2)+2 (b^2) (c^2)-(c^4)+2 (a^2) 𝒵]) ),(1/(b ))(2Area+(Sqrt[-(a^4)+(a^2) (b^2)-2 (b^4)+2 (a^2) (c^2)+(b^2) (c^2)-(c^4)+2 (b^2) 𝒵]) ),((2Area-(Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+(a^2) (c^2)+(b^2) (c^2)-2 (c^4)+2 (c^2) 𝒵]))/c)};

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

Cartesian[ToSidelengths[expr6/.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)])]

ToAngles[-(a^4)+(a^2) (b^2)-2 (b^4)+2 (a^2) (c^2)+(b^2) (c^2)-(c^4)]

bickart=FullSimplify[ToAngles/@{expr5,expr6},TriangleQ[]&&𝒵>0]

bickart={{(((Sqrt[2]) Area)/a)- (Sqrt[-(a^2)+𝒵+b c Cos[B-C]]),(((Sqrt[2]) Area)/b)+ (Sqrt[-(b^2)+𝒵+a c Cos[A-C]]),(((Sqrt[2]) Area)/c)-(Sqrt[-(c^2)+𝒵+a b Cos[A-B]])},{(((Sqrt[2]) Area)/a)+ (Sqrt[-(a^2)+𝒵+b c Cos[B-C]]),(((Sqrt[2]) Area)/b)- (Sqrt[-(b^2)+𝒵+a c Cos[A-C]]),(((Sqrt[2]) Area)/c)+ (Sqrt[-(c^2)+𝒵+a b Cos[A-B]])}};

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