Second Steiner Circle

Author

Eric W. Weisstein
January 2, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{(3+1/6)/10,5/10}}]//N[#,10]&;
Show[Graphics[
{
{Purple,Sides[t,2]},
t,
{Blue,
(Cartesian[#[t]]&/@(Cevians[SteinerPoint]/.Line[x__]:>Line[x,Trilinears[SteinerPoint]]))/.Line:>LongestLine
},
{Green,SteinerTriangle[t]},
{Red,SecondSteinerCircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[SteinerTriangle[t]],{"(Subscript[S, A])","(Subscript[S, B])","(Subscript[S, C])"}}],
{
{SteinerPoint[t],"S",Blue}
}
],LabelOffset->.06,PointSize[.025]]
}
],AspectRatio->Automatic,PlotRange->{{-.2,2.2},{-.2,1.5}}]

Incident Centers

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

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

t=Triangle[{{0,0},{1,0},{(3+2/3)/10,5/10}}]//N[#,10]&;
Show[Graphics[{
t,
{Red,SecondSteinerCircle[t]},
{Green,SteinerTriangle[t]},
{Blue,NinePointCircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[SteinerTriangle[t]],{"(Subscript[S, A])","(Subscript[S, B])","(Subscript[S, C])"}}],
{
{X[114][t],"(Subscript[X, 114])",Purple},
{Cartesian[CyclicTrilinears[(b^2-c^2)^2 (2 a^2-b^2-c^2) (a^4-b^4-c^4+b^2 c^2)/a][t]],"P",Purple}
}
],LabelOffset->.06,PointSize[.02]]
}],AspectRatio->Automatic]

Circle Center

Center[SecondSteinerCircle][[1]]//TraditionalForm

Center[SecondSteinerCircle][[1]]//TeXForm

Circle Radius

CircleRadius[SecondSteinerCircle]//TraditionalForm

Circle Function

CircleFunction[SecondSteinerCircle]//TraditionalForm

Derivation

circ1=Together//@Circumcircle@@TrilinearVertexMatrix[SteinerTriangle];

IncidentCenters[KimberlingCenters,Center[circ1]]//Timing

circ=circ1/.Circle[ctr_List,r_]:>Circle[Symmetrize/@ctr,Simplify[Together//@r]]

circ=circ1/.Circle[ctr_List,r_]:>Circle[Symmetrize/@ctr,Together//@ExpandAll[Together//@r]]

Center[circ][[1]]//InputForm

r=CircleRadius[circ];

sqrt=Factor[((a^16)+6 (a^15) c+(a^14) (-4 (b^2)+13 (c^2))+(a^13) (-26 (b^2) c+14 (c^3))+((((b^2)-(c^2)))^4) ((((b^4)+(c^4)))^2)+(a^12) (8 (b^4)-56 (b^2) (c^2)+6 (c^4))+2 (a^11) (25 (b^4) c-24 (b^2) (c^3)-7 (c^5))-2 a c ((((b^2)-(c^2)))^4) ((b^6)-3 (b^4) (c^2)+(b^2) (c^4)-3 (c^6))-(a^10) (12 (b^6)-101 (b^4) (c^2)+2 (b^2) (c^4)+29 (c^6))+(a^9) (-62 (b^6) c+86 (b^4) (c^3)+30 (b^2) (c^5)-6 (c^7))+(a^8) (14 (b^8)-120 (b^6) (c^2)+31 (b^4) (c^4)+30 (b^2) (c^6)+18 (c^8))+2 (a^7) (29 (b^8) c-60 (b^6) (c^3)+(b^4) (c^5)+12 (b^2) (c^7)-3 (c^9))-(a^2) ((((b^2)-(c^2)))^2) (4 (b^10)-19 (b^8) (c^2)+34 (b^6) (c^4)-28 (b^4) (c^6)+30 (b^2) (c^8)-13 (c^10))-(a^6) (12 (b^10)-115 (b^8) (c^2)+104 (b^6) (c^4)+3 (b^4) (c^6)-30 (b^2) (c^8)+29 (c^10))-2 (a^5) (19 (b^10) c-67 (b^8) (c^3)+51 (b^6) (c^5)-(b^4) (c^7)-15 (b^2) (c^9)+7 (c^11))+(a^4) (8 (b^12)-76 (b^10) (c^2)+141 (b^8) (c^4)-104 (b^6) (c^6)+31 (b^4) (c^8)-2 (b^2) (c^10)+6 (c^12))+2 (a^3) (7 (b^12) c-40 (b^10) (c^3)+67 (b^8) (c^5)-60 (b^6) (c^7)+43 (b^4) (c^9)-24 (b^2) (c^11)+7 (c^13)))]

Factor[r^2/.(Sqrt[(b^2) sqrt]) ->b f]

expr=((a^16)-4 (a^14) (b^2)+8 (a^12) (b^4)-12 (a^10) (b^6)+14 (a^8) (b^8)-12 (a^6) (b^10)+8 (a^4) (b^12)-4 (a^2) (b^14)+(b^16)+6 (a^15) c-22 (a^13) (b^2) c+38 (a^11) (b^4) c-46 (a^9) (b^6) c+42 (a^7) (b^8) c-26 (a^5) (b^10) c+10 (a^3) (b^12) c-2 a (b^14) c+13 (a^14) (c^2)-40 (a^12) (b^2) (c^2)+61 (a^10) (b^4) (c^2)-72 (a^8) (b^6) (c^2)+67 (a^6) (b^8) (c^2)-44 (a^4) (b^10) (c^2)+19 (a^2) (b^12) (c^2)-4 (b^14) (c^2)+10 (a^13) (c^3)-16 (a^11) (b^2) (c^3)+30 (a^9) (b^4) (c^3)-56 (a^7) (b^6) (c^3)+62 (a^5) (b^8) (c^3)-40 (a^3) (b^10) (c^3)+10 a (b^12) (c^3)-10 (a^12) (c^4)+38 (a^10) (b^2) (c^4)-9 (a^8) (b^4) (c^4)-40 (a^6) (b^6) (c^4)+61 (a^4) (b^8) (c^4)-44 (a^2) (b^10) (c^4)+8 (b^12) (c^4)-34 (a^11) (c^5)+46 (a^9) (b^2) (c^5)+2 (a^7) (b^4) (c^5)-38 (a^5) (b^6) (c^5)+62 (a^3) (b^8) (c^5)-26 a (b^10) (c^5)-29 (a^10) (c^6)-10 (a^8) (b^2) (c^6)+21 (a^6) (b^4) (c^6)-40 (a^4) (b^6) (c^6)+67 (a^2) (b^8) (c^6)-12 (b^10) (c^6)+18 (a^9) (c^7)-48 (a^7) (b^2) (c^7)+2 (a^5) (b^4) (c^7)-56 (a^3) (b^6) (c^7)+42 a (b^8) (c^7)+50 (a^8) (c^8)-10 (a^6) (b^2) (c^8)-9 (a^4) (b^4) (c^8)-72 (a^2) (b^6) (c^8)+14 (b^8) (c^8)+18 (a^7) (c^9)+46 (a^5) (b^2) (c^9)+30 (a^3) (b^4) (c^9)-46 a (b^6) (c^9)-29 (a^6) (c^10)+38 (a^4) (b^2) (c^10)+61 (a^2) (b^4) (c^10)-12 (b^6) (c^10)-34 (a^5) (c^11)-16 (a^3) (b^2) (c^11)+38 a (b^4) (c^11)-10 (a^4) (c^12)-40 (a^2) (b^2) (c^12)+8 (b^4) (c^12)+10 (a^3) (c^13)-22 a (b^2) (c^13)+13 (a^2) (c^14)-4 (b^2) (c^14)+6 a (c^15)+(c^16)-(a^8) f+2 (a^6) (b^2) f-2 (a^4) (b^4) f+2 (a^2) (b^6) f-(b^8) f-3 (a^7) c f+3 (a^5) (b^2) c f-(a^3) (b^4) c f+a (b^6) c f-2 (a^6) (c^2) f-(a^4) (b^2) (c^2) f-(a^2) (b^4) (c^2) f+2 (b^6) (c^2) f+3 (a^5) (c^3) f-5 (a^3) (b^2) (c^3) f-a (b^4) (c^3) f+6 (a^4) (c^4) f-(a^2) (b^2) (c^4) f-2 (b^4) (c^4) f+3 (a^3) (c^5) f+3 a (b^2) (c^5) f-2 (a^2) (c^6) f+2 (b^2) (c^6) f-3 a (c^7) f-(c^8) f);

term=Factor[Coefficient[expr,f]]

rest=Factor[expr-term f]

((-(a^6)+3 (a^4) (b^2)-3 (a^2) (b^4)+(b^6)-(a^4) (c^2)+3 (a^2) (b^2) (c^2)-3 (b^4) (c^2)-(a^2) (c^4)+3 (b^2) (c^4)-(c^6)) ((a^6)+(a^4) (b^2)+(a^2) (b^4)+(b^6)-3 (a^4) (c^2)-3 (a^2) (b^2) (c^2)-3 (b^4) (c^2)+3 (a^2) (c^4)+3 (b^2) (c^4)-(c^6)) (-(a^6)+3 (a^4) (b^2)-3 (a^2) (b^4)+(b^6)+3 (a^4) (c^2)-3 (a^2) (b^2) (c^2)+(b^4) (c^2)-3 (a^2) (c^4)+(b^2) (c^4)+(c^6)) (term f+rest))/(2 (((-a+b))^2) (((a+b))^2) (((a-c))^2) (((b-c))^2) (-a+b-c) (a+b-c) (((a+c))^2) (((b+c))^2) (-a+b+c) (a+b+c) ((((a^8)-2 (a^6) (b^2)+2 (a^4) (b^4)-2 (a^2) (b^6)+(b^8)+3 (a^7) c-3 (a^5) (b^2) c+(a^3) (b^4) c-a (b^6) c+2 (a^6) (c^2)+(a^4) (b^2) (c^2)+(a^2) (b^4) (c^2)-2 (b^6) (c^2)-3 (a^5) (c^3)+5 (a^3) (b^2) (c^3)+a (b^4) (c^3)-6 (a^4) (c^4)+(a^2) (b^2) (c^4)+2 (b^4) (c^4)-3 (a^3) (c^5)-3 a (b^2) (c^5)+2 (a^2) (c^6)-2 (b^2) (c^6)+3 a (c^7)+(c^8)-f))^2))

((-(a^6)+3 (a^4) (b^2)-3 (a^2) (b^4)+(b^6)-(a^4) (c^2)+3 (a^2) (b^2) (c^2)-3 (b^4) (c^2)-(a^2) (c^4)+3 (b^2) (c^4)-(c^6)) ((a^6)+(a^4) (b^2)+(a^2) (b^4)+(b^6)-3 (a^4) (c^2)-3 (a^2) (b^2) (c^2)-3 (b^4) (c^2)+3 (a^2) (c^4)+3 (b^2) (c^4)-(c^6)) (-(a^6)+3 (a^4) (b^2)-3 (a^2) (b^4)+(b^6)+3 (a^4) (c^2)-3 (a^2) (b^2) (c^2)+(b^4) (c^2)-3 (a^2) (c^4)+(b^2) (c^4)+(c^6)) ((a^16)-4 (a^14) (b^2)+8 (a^12) (b^4)-12 (a^10) (b^6)+14 (a^8) (b^8)-12 (a^6) (b^10)+8 (a^4) (b^12)-4 (a^2) (b^14)+(b^16)+6 (a^15) c-22 (a^13) (b^2) c+38 (a^11) (b^4) c-46 (a^9) (b^6) c+42 (a^7) (b^8) c-26 (a^5) (b^10) c+10 (a^3) (b^12) c-2 a (b^14) c+13 (a^14) (c^2)-40 (a^12) (b^2) (c^2)+61 (a^10) (b^4) (c^2)-72 (a^8) (b^6) (c^2)+67 (a^6) (b^8) (c^2)-44 (a^4) (b^10) (c^2)+19 (a^2) (b^12) (c^2)-4 (b^14) (c^2)+10 (a^13) (c^3)-16 (a^11) (b^2) (c^3)+30 (a^9) (b^4) (c^3)-56 (a^7) (b^6) (c^3)+62 (a^5) (b^8) (c^3)-40 (a^3) (b^10) (c^3)+10 a (b^12) (c^3)-10 (a^12) (c^4)+38 (a^10) (b^2) (c^4)-9 (a^8) (b^4) (c^4)-40 (a^6) (b^6) (c^4)+61 (a^4) (b^8) (c^4)-44 (a^2) (b^10) (c^4)+8 (b^12) (c^4)-34 (a^11) (c^5)+46 (a^9) (b^2) (c^5)+2 (a^7) (b^4) (c^5)-38 (a^5) (b^6) (c^5)+62 (a^3) (b^8) (c^5)-26 a (b^10) (c^5)-29 (a^10) (c^6)-10 (a^8) (b^2) (c^6)+21 (a^6) (b^4) (c^6)-40 (a^4) (b^6) (c^6)+67 (a^2) (b^8) (c^6)-12 (b^10) (c^6)+18 (a^9) (c^7)-48 (a^7) (b^2) (c^7)+2 (a^5) (b^4) (c^7)-56 (a^3) (b^6) (c^7)+42 a (b^8) (c^7)+50 (a^8) (c^8)-10 (a^6) (b^2) (c^8)-9 (a^4) (b^4) (c^8)-72 (a^2) (b^6) (c^8)+14 (b^8) (c^8)+18 (a^7) (c^9)+46 (a^5) (b^2) (c^9)+30 (a^3) (b^4) (c^9)-46 a (b^6) (c^9)-29 (a^6) (c^10)+38 (a^4) (b^2) (c^10)+61 (a^2) (b^4) (c^10)-12 (b^6) (c^10)-34 (a^5) (c^11)-16 (a^3) (b^2) (c^11)+38 a (b^4) (c^11)-10 (a^4) (c^12)-40 (a^2) (b^2) (c^12)+8 (b^4) (c^12)+10 (a^3) (c^13)-22 a (b^2) (c^13)+13 (a^2) (c^14)-4 (b^2) (c^14)+6 a (c^15)+(c^16)-((a^2)-(b^2)+a c+(c^2)) ((a^6)-(a^4) (b^2)+(a^2) (b^4)-(b^6)+2 (a^5) c-(a^4) (c^2)+(a^2) (b^2) (c^2)+(b^4) (c^2)-4 (a^3) (c^3)-(a^2) (c^4)-(b^2) (c^4)+2 a (c^5)+(c^6)) f))/(2 (((-a+b))^2) (((a+b))^2) (((a-c))^2) (((b-c))^2) (-a+b-c) (a+b-c) (((a+c))^2) (((b+c))^2) (-a+b+c) (a+b+c) (((Factor[(a^8)-2 (a^6) (b^2)+2 (a^4) (b^4)-2 (a^2) (b^6)+(b^8)+3 (a^7) c-3 (a^5) (b^2) c+(a^3) (b^4) c-a (b^6) c+2 (a^6) (c^2)+(a^4) (b^2) (c^2)+(a^2) (b^4) (c^2)-2 (b^6) (c^2)-3 (a^5) (c^3)+5 (a^3) (b^2) (c^3)+a (b^4) (c^3)-6 (a^4) (c^4)+(a^2) (b^2) (c^4)+2 (b^4) (c^4)-3 (a^3) (c^5)-3 a (b^2) (c^5)+2 (a^2) (c^6)-2 (b^2) (c^6)+3 a (c^7)+(c^8)]-f))^2))

%/.((a^6)-(a^4) (b^2)+(a^2) (b^4)-(b^6)+2 (a^5) c-(a^4) (c^2)+(a^2) (b^2) (c^2)+(b^4) (c^2)-4 (a^3) (c^3)-(a^2) (c^4)-(b^2) (c^4)+2 a (c^5)+(c^6))->g

Factor[sqrt]

(rr=With[{
g=((a^6)-(a^4) (b^2)+(a^2) (b^4)-(b^6)+2 (a^5) c-(a^4) (c^2)+(a^2) (b^2) (c^2)+(b^4) (c^2)-4 (a^3) (c^3)-(a^2) (c^4)-(b^2) (c^4)+2 a (c^5)+(c^6)),f=(Sqrt[sqrt])
},
1/(Abs[f-((a^2)-(b^2)+a c+(c^2)) g](a+b)(b+c)(c+a)Abs[(a-b)(b-c)(c-a)])√-(((-(a^6)+3 (a^4) (b^2)-3 (a^2) (b^4)+(b^6)-(a^4) (c^2)+3 (a^2) (b^2) (c^2)-3 (b^4) (c^2)-(a^2) (c^4)+3 (b^2) (c^4)-(c^6)) ((a^6)+(a^4) (b^2)+(a^2) (b^4)+(b^6)-3 (a^4) (c^2)-3 (a^2) (b^2) (c^2)-3 (b^4) (c^2)+3 (a^2) (c^4)+3 (b^2) (c^4)-(c^6)) (-(a^6)+3 (a^4) (b^2)-3 (a^2) (b^4)+(b^6)+3 (a^4) (c^2)-3 (a^2) (b^2) (c^2)+(b^4) (c^2)-3 (a^2) (c^4)+(b^2) (c^4)+(c^6)) ((a^16)-4 (a^14) (b^2)+8 (a^12) (b^4)-12 (a^10) (b^6)+14 (a^8) (b^8)-12 (a^6) (b^10)+8 (a^4) (b^12)-4 (a^2) (b^14)+(b^16)+6 (a^15) c-22 (a^13) (b^2) c+38 (a^11) (b^4) c-46 (a^9) (b^6) c+42 (a^7) (b^8) c-26 (a^5) (b^10) c+10 (a^3) (b^12) c-2 a (b^14) c+13 (a^14) (c^2)-40 (a^12) (b^2) (c^2)+61 (a^10) (b^4) (c^2)-72 (a^8) (b^6) (c^2)+67 (a^6) (b^8) (c^2)-44 (a^4) (b^10) (c^2)+19 (a^2) (b^12) (c^2)-4 (b^14) (c^2)+10 (a^13) (c^3)-16 (a^11) (b^2) (c^3)+30 (a^9) (b^4) (c^3)-56 (a^7) (b^6) (c^3)+62 (a^5) (b^8) (c^3)-40 (a^3) (b^10) (c^3)+10 a (b^12) (c^3)-10 (a^12) (c^4)+38 (a^10) (b^2) (c^4)-9 (a^8) (b^4) (c^4)-40 (a^6) (b^6) (c^4)+61 (a^4) (b^8) (c^4)-44 (a^2) (b^10) (c^4)+8 (b^12) (c^4)-34 (a^11) (c^5)+46 (a^9) (b^2) (c^5)+2 (a^7) (b^4) (c^5)-38 (a^5) (b^6) (c^5)+62 (a^3) (b^8) (c^5)-26 a (b^10) (c^5)-29 (a^10) (c^6)-10 (a^8) (b^2) (c^6)+21 (a^6) (b^4) (c^6)-40 (a^4) (b^6) (c^6)+67 (a^2) (b^8) (c^6)-12 (b^10) (c^6)+18 (a^9) (c^7)-48 (a^7) (b^2) (c^7)+2 (a^5) (b^4) (c^7)-56 (a^3) (b^6) (c^7)+42 a (b^8) (c^7)+50 (a^8) (c^8)-10 (a^6) (b^2) (c^8)-9 (a^4) (b^4) (c^8)-72 (a^2) (b^6) (c^8)+14 (b^8) (c^8)+18 (a^7) (c^9)+46 (a^5) (b^2) (c^9)+30 (a^3) (b^4) (c^9)-46 a (b^6) (c^9)-29 (a^6) (c^10)+38 (a^4) (b^2) (c^10)+61 (a^2) (b^4) (c^10)-12 (b^6) (c^10)-34 (a^5) (c^11)-16 (a^3) (b^2) (c^11)+38 a (b^4) (c^11)-10 (a^4) (c^12)-40 (a^2) (b^2) (c^12)+8 (b^4) (c^12)+10 (a^3) (c^13)-22 a (b^2) (c^13)+13 (a^2) (c^14)-4 (b^2) (c^14)+6 a (c^15)+(c^16)-((a^2)-(b^2)+a c+(c^2)) f g))/(2  (a-b+c) (a+b-c) (-a+b+c) (a+b+c) ))
]);

t=Triangle[{{0,0},{1,0},{3,5}/10}]//N[#,20]&;

r/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

rr/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

InputForm[rr]

Map[Factor,Together//@ExpandAll[Together//@CircleRadius[SecondSteinerCircle]],3]

...big mess...

r2=Factor[Together//@ExpandAll[Together//@ToSidelengths[CircleRadius[Numerator[Factor[TrilinearEquation[SecondSteinerCircle]]]]^2]]]

CircleRadius[SecondSteinerCircle]//TraditionalForm

r=Sqrt[-(((-a^6+3*b^2*a^4-c^2*a^4-3*b^4*a^2-c^4*a^2+3*b^2*c^2*a^2+b^6-c^6+3*b^2*c^4-3*b^4*c^2)*(a^6+b^2*a^4-3*c^2*a^4+b^4*a^2+3*c^4*a^2-3*b^2*c^2*a^2+b^6-c^6+3*b^2*c^4-3*b^4*c^2)*(-a^6+3*b^2*a^4+3*c^2*a^4-3*b^4*a^2-3*c^4*a^2-3*b^2*c^2*a^2+b^6+c^6+b^2*c^4+b^4*c^2)*(a^16+6*c*a^15-4*b^2*a^14+13*c^2*a^14+10*c^3*a^13-22*b^2*c*a^13+8*b^4*a^12-10*c^4*a^12-40*b^2*c^2*a^12-34*c^5*a^11-16*b^2*c^3*a^11+38*b^4*c*a^11-12*b^6*a^10-29*c^6*a^10+38*b^2*c^4*a^10+61*b^4*c^2*a^10+18*c^7*a^9+46*b^2*c^5*a^9+30*b^4*c^3*a^9-46*b^6*c*a^9+14*b^8*a^8+50*c^8*a^8-10*b^2*c^6*a^8-9*b^4*c^4*a^8-72*b^6*c^2*a^8+18*c^9*a^7-48*b^2*c^7*a^7+2*b^4*c^5*a^7-56*b^6*c^3*a^7+42*b^8*c*a^7-12*b^10*a^6-29*c^10*a^6-10*b^2*c^8*a^6+21*b^4*c^6*a^6-40*b^6*c^4*a^6+67*b^8*c^2*a^6-34*c^11*a^5+46*b^2*c^9*a^5+2*b^4*c^7*a^5-38*b^6*c^5*a^5+62*b^8*c^3*a^5-26*b^10*c*a^5+8*b^12*a^4-10*c^12*a^4+38*b^2*c^10*a^4-9*b^4*c^8*a^4-40*b^6*c^6*a^4+61*b^8*c^4*a^4-44*b^10*c^2*a^4+10*c^13*a^3-16*b^2*c^11*a^3+30*b^4*c^9*a^3-56*b^6*c^7*a^3+62*b^8*c^5*a^3-40*b^10*c^3*a^3+10*b^12*c*a^3-4*b^14*a^2+13*c^14*a^2-40*b^2*c^12*a^2+61*b^4*c^10*a^2-72*b^6*c^8*a^2+67*b^8*c^6*a^2-44*b^10*c^4*a^2+19*b^12*c^2*a^2+6*c^15*a-22*b^2*c^13*a+38*b^4*c^11*a-46*b^6*c^9*a+42*b^8*c^7*a-26*b^10*c^5*a+10*b^12*c^3*a-2*b^14*c*a+b^16+c^16-4*b^2*c^14+8*b^4*c^12-12*b^6*c^10+14*b^8*c^8-12*b^10*c^6+8*b^12*c^4-4*b^14*c^2-(a^2+c*a-b^2+c^2)*(a^6+2*c*a^5-b^2*a^4-c^2*a^4-4*c^3*a^3+b^4*a^2-c^4*a^2+b^2*c^2*a^2+2*c^5*a-b^6+c^6-b^2*c^4+b^4*c^2)*Sqrt[a^16+6*c*a^15-4*b^2*a^14+13*c^2*a^14+14*c^3*a^13-26*b^2*c*a^13+8*b^4*a^12+6*c^4*a^12-56*b^2*c^2*a^12-14*c^5*a^11-48*b^2*c^3*a^11+50*b^4*c*a^11-12*b^6*a^10-29*c^6*a^10-2*b^2*c^4*a^10+101*b^4*c^2*a^10-6*c^7*a^9+30*b^2*c^5*a^9+86*b^4*c^3*a^9-62*b^6*c*a^9+14*b^8*a^8+18*c^8*a^8+30*b^2*c^6*a^8+31*b^4*c^4*a^8-120*b^6*c^2*a^8-6*c^9*a^7+24*b^2*c^7*a^7+2*b^4*c^5*a^7-120*b^6*c^3*a^7+58*b^8*c*a^7-12*b^10*a^6-29*c^10*a^6+30*b^2*c^8*a^6-3*b^4*c^6*a^6-104*b^6*c^4*a^6+115*b^8*c^2*a^6-14*c^11*a^5+30*b^2*c^9*a^5+2*b^4*c^7*a^5-102*b^6*c^5*a^5+134*b^8*c^3*a^5-38*b^10*c*a^5+8*b^12*a^4+6*c^12*a^4-2*b^2*c^10*a^4+31*b^4*c^8*a^4-104*b^6*c^6*a^4+141*b^8*c^4*a^4-76*b^10*c^2*a^4+14*c^13*a^3-48*b^2*c^11*a^3+86*b^4*c^9*a^3-120*b^6*c^7*a^3+134*b^8*c^5*a^3-80*b^10*c^3*a^3+14*b^12*c*a^3-4*b^14*a^2+13*c^14*a^2-56*b^2*c^12*a^2+101*b^4*c^10*a^2-120*b^6*c^8*a^2+115*b^8*c^6*a^2-76*b^10*c^4*a^2+27*b^12*c^2*a^2+6*c^15*a-26*b^2*c^13*a+50*b^4*c^11*a-62*b^6*c^9*a+58*b^8*c^7*a-38*b^10*c^5*a+14*b^12*c^3*a-2*b^14*c*a+b^16+c^16-4*b^2*c^14+8*b^4*c^12-12*b^6*c^10+14*b^8*c^8-12*b^10*c^6+8*b^12*c^4-4*b^14*c^2]))/((a+b-c)*(a-b+c)*(-a+b+c)*(a+b+c)))]/(Sqrt[2]*(a+b)*(a+c)*(b+c)*Abs[(a-b)*(b-c)*(c-a)]*Abs[Sqrt[a^16+6*c*a^15-4*b^2*a^14+13*c^2*a^14+14*c^3*a^13-26*b^2*c*a^13+8*b^4*a^12+6*c^4*a^12-56*b^2*c^2*a^12-14*c^5*a^11-48*b^2*c^3*a^11+50*b^4*c*a^11-12*b^6*a^10-29*c^6*a^10-2*b^2*c^4*a^10+101*b^4*c^2*a^10-6*c^7*a^9+30*b^2*c^5*a^9+86*b^4*c^3*a^9-62*b^6*c*a^9+14*b^8*a^8+18*c^8*a^8+30*b^2*c^6*a^8+31*b^4*c^4*a^8-120*b^6*c^2*a^8-6*c^9*a^7+24*b^2*c^7*a^7+2*b^4*c^5*a^7-120*b^6*c^3*a^7+58*b^8*c*a^7-12*b^10*a^6-29*c^10*a^6+30*b^2*c^8*a^6-3*b^4*c^6*a^6-104*b^6*c^4*a^6+115*b^8*c^2*a^6-14*c^11*a^5+30*b^2*c^9*a^5+2*b^4*c^7*a^5-102*b^6*c^5*a^5+134*b^8*c^3*a^5-38*b^10*c*a^5+8*b^12*a^4+6*c^12*a^4-2*b^2*c^10*a^4+31*b^4*c^8*a^4-104*b^6*c^6*a^4+141*b^8*c^4*a^4-76*b^10*c^2*a^4+14*c^13*a^3-48*b^2*c^11*a^3+86*b^4*c^9*a^3-120*b^6*c^7*a^3+134*b^8*c^5*a^3-80*b^10*c^3*a^3+14*b^12*c*a^3-4*b^14*a^2+13*c^14*a^2-56*b^2*c^12*a^2+101*b^4*c^10*a^2-120*b^6*c^8*a^2+115*b^8*c^6*a^2-76*b^10*c^4*a^2+27*b^12*c^2*a^2+6*c^15*a-26*b^2*c^13*a+50*b^4*c^11*a-62*b^6*c^9*a+58*b^8*c^7*a-38*b^10*c^5*a+14*b^12*c^3*a-2*b^14*c*a+b^16+c^16-4*b^2*c^14+8*b^4*c^12-12*b^6*c^10+14*b^8*c^8-12*b^10*c^6+8*b^12*c^4-4*b^14*c^2]-(a^2+c*a-b^2+c^2)*(a^6+2*c*a^5-b^2*a^4-c^2*a^4-4*c^3*a^3+b^4*a^2-c^4*a^2+b^2*c^2*a^2+2*c^5*a-b^6+c^6-b^2*c^4+b^4*c^2)]);

Simplify[r,TriangleQ[]]//Timing

Aborted after running overnight

t=Triangle[{{0,0},{1,0},{3/10,5/10}}]//N[#,20]&;
CircleRadius[SecondSteinerCircle]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

Sqrt[(Times@@CyclicTrilinears [(b^6 + c^6 - a^6 + 3 a^4 b^2 - 3 a^2 b^4 + b^4 c^2 + b^2
c^4 - 3 c^4 a^2 + 3 a^4 c^2 - 3 a^2 b^2 c^2) / (b^2 - c^2)^2] / (64 Area^2)//ToSidelengths)]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

Factor[CircleRadius[SecondSteinerCircle]^2/.((a^16)-4 (a^14) (b^2)+8 (a^12) (b^4)-12 (a^10) (b^6)+14 (a^8) (b^8)-12 (a^6) (b^10)+8 (a^4) (b^12)-4 (a^2) (b^14)+(b^16)+6 (a^15) c-26 (a^13) (b^2) c+50 (a^11) (b^4) c-62 (a^9) (b^6) c+58 (a^7) (b^8) c-38 (a^5) (b^10) c+14 (a^3) (b^12) c-2 a (b^14) c+13 (a^14) (c^2)-56 (a^12) (b^2) (c^2)+101 (a^10) (b^4) (c^2)-120 (a^8) (b^6) (c^2)+115 (a^6) (b^8) (c^2)-76 (a^4) (b^10) (c^2)+27 (a^2) (b^12) (c^2)-4 (b^14) (c^2)+14 (a^13) (c^3)-48 (a^11) (b^2) (c^3)+86 (a^9) (b^4) (c^3)-120 (a^7) (b^6) (c^3)+134 (a^5) (b^8) (c^3)-80 (a^3) (b^10) (c^3)+14 a (b^12) (c^3)+6 (a^12) (c^4)-2 (a^10) (b^2) (c^4)+31 (a^8) (b^4) (c^4)-104 (a^6) (b^6) (c^4)+141 (a^4) (b^8) (c^4)-76 (a^2) (b^10) (c^4)+8 (b^12) (c^4)-14 (a^11) (c^5)+30 (a^9) (b^2) (c^5)+2 (a^7) (b^4) (c^5)-102 (a^5) (b^6) (c^5)+134 (a^3) (b^8) (c^5)-38 a (b^10) (c^5)-29 (a^10) (c^6)+30 (a^8) (b^2) (c^6)-3 (a^6) (b^4) (c^6)-104 (a^4) (b^6) (c^6)+115 (a^2) (b^8) (c^6)-12 (b^10) (c^6)-6 (a^9) (c^7)+24 (a^7) (b^2) (c^7)+2 (a^5) (b^4) (c^7)-120 (a^3) (b^6) (c^7)+58 a (b^8) (c^7)+18 (a^8) (c^8)+30 (a^6) (b^2) (c^8)+31 (a^4) (b^4) (c^8)-120 (a^2) (b^6) (c^8)+14 (b^8) (c^8)-6 (a^7) (c^9)+30 (a^5) (b^2) (c^9)+86 (a^3) (b^4) (c^9)-62 a (b^6) (c^9)-29 (a^6) (c^10)-2 (a^4) (b^2) (c^10)+101 (a^2) (b^4) (c^10)-12 (b^6) (c^10)-14 (a^5) (c^11)-48 (a^3) (b^2) (c^11)+50 a (b^4) (c^11)+6 (a^4) (c^12)-56 (a^2) (b^2) (c^12)+8 (b^4) (c^12)+14 (a^3) (c^13)-26 a (b^2) (c^13)+13 (a^2) (c^14)-4 (b^2) (c^14)+6 a (c^15)+(c^16))->poly]/. 1/Abs[x_]^2:>1/x^2

f=Factor[Coefficient[ll=(a^16)-4 (a^14) (b^2)+8 (a^12) (b^4)-12 (a^10) (b^6)+14 (a^8) (b^8)-12 (a^6) (b^10)+8 (a^4) (b^12)-4 (a^2) (b^14)+(b^16)+6 (a^15) c-22 (a^13) (b^2) c+38 (a^11) (b^4) c-46 (a^9) (b^6) c+42 (a^7) (b^8) c-26 (a^5) (b^10) c+10 (a^3) (b^12) c-2 a (b^14) c+13 (a^14) (c^2)-40 (a^12) (b^2) (c^2)+61 (a^10) (b^4) (c^2)-72 (a^8) (b^6) (c^2)+67 (a^6) (b^8) (c^2)-44 (a^4) (b^10) (c^2)+19 (a^2) (b^12) (c^2)-4 (b^14) (c^2)+10 (a^13) (c^3)-16 (a^11) (b^2) (c^3)+30 (a^9) (b^4) (c^3)-56 (a^7) (b^6) (c^3)+62 (a^5) (b^8) (c^3)-40 (a^3) (b^10) (c^3)+10 a (b^12) (c^3)-10 (a^12) (c^4)+38 (a^10) (b^2) (c^4)-9 (a^8) (b^4) (c^4)-40 (a^6) (b^6) (c^4)+61 (a^4) (b^8) (c^4)-44 (a^2) (b^10) (c^4)+8 (b^12) (c^4)-34 (a^11) (c^5)+46 (a^9) (b^2) (c^5)+2 (a^7) (b^4) (c^5)-38 (a^5) (b^6) (c^5)+62 (a^3) (b^8) (c^5)-26 a (b^10) (c^5)-29 (a^10) (c^6)-10 (a^8) (b^2) (c^6)+21 (a^6) (b^4) (c^6)-40 (a^4) (b^6) (c^6)+67 (a^2) (b^8) (c^6)-12 (b^10) (c^6)+18 (a^9) (c^7)-48 (a^7) (b^2) (c^7)+2 (a^5) (b^4) (c^7)-56 (a^3) (b^6) (c^7)+42 a (b^8) (c^7)+50 (a^8) (c^8)-10 (a^6) (b^2) (c^8)-9 (a^4) (b^4) (c^8)-72 (a^2) (b^6) (c^8)+14 (b^8) (c^8)+18 (a^7) (c^9)+46 (a^5) (b^2) (c^9)+30 (a^3) (b^4) (c^9)-46 a (b^6) (c^9)-29 (a^6) (c^10)+38 (a^4) (b^2) (c^10)+61 (a^2) (b^4) (c^10)-12 (b^6) (c^10)-34 (a^5) (c^11)-16 (a^3) (b^2) (c^11)+38 a (b^4) (c^11)-10 (a^4) (c^12)-40 (a^2) (b^2) (c^12)+8 (b^4) (c^12)+10 (a^3) (c^13)-22 a (b^2) (c^13)+13 (a^2) (c^14)-4 (b^2) (c^14)+6 a (c^15)+(c^16)-(a^8) (Sqrt[poly])+2 (a^6) (b^2) (Sqrt[poly])-2 (a^4) (b^4) (Sqrt[poly])+2 (a^2) (b^6) (Sqrt[poly])-(b^8) (Sqrt[poly])-3 (a^7) c (Sqrt[poly])+3 (a^5) (b^2) c (Sqrt[poly])-(a^3) (b^4) c (Sqrt[poly])+a (b^6) c (Sqrt[poly])-2 (a^6) (c^2) (Sqrt[poly])-(a^4) (b^2) (c^2) (Sqrt[poly])-(a^2) (b^4) (c^2) (Sqrt[poly])+2 (b^6) (c^2) (Sqrt[poly])+3 (a^5) (c^3) (Sqrt[poly])-5 (a^3) (b^2) (c^3) (Sqrt[poly])-a (b^4) (c^3) (Sqrt[poly])+6 (a^4) (c^4) (Sqrt[poly])-(a^2) (b^2) (c^4) (Sqrt[poly])-2 (b^4) (c^4) (Sqrt[poly])+3 (a^3) (c^5) (Sqrt[poly])+3 a (b^2) (c^5) (Sqrt[poly])-2 (a^2) (c^6) (Sqrt[poly])+2 (b^2) (c^6) (Sqrt[poly])-3 a (c^7) (Sqrt[poly])-(c^8) (Sqrt[poly]),(Sqrt[poly])]]

Factor[ll-f(Sqrt[poly])]

n=((a^6)-3 (a^4) (b^2)+3 (a^2) (b^4)-(b^6)-3 (a^4) (c^2)+3 (a^2) (b^2) (c^2)-(b^4) (c^2)+3 (a^2) (c^4)-(b^2) (c^4)-(c^6)) ((a^6)+(a^4) (b^2)+(a^2) (b^4)+(b^6)-3 (a^4) (c^2)-3 (a^2) (b^2) (c^2)-3 (b^4) (c^2)+3 (a^2) (c^4)+3 (b^2) (c^4)-(c^6)) ((a^6)-3 (a^4) (b^2)+3 (a^2) (b^4)-(b^6)+(a^4) (c^2)-3 (a^2) (b^2) (c^2)+3 (b^4) (c^2)+(a^2) (c^4)-3 (b^2) (c^4)+(c^6)) (poly2-((a^2)-(b^2)+a c+(c^2)) ((a^6)-(a^4) (b^2)+(a^2) (b^4)-(b^6)+2 (a^5) c-(a^4) (c^2)+(a^2) (b^2) (c^2)+(b^4) (c^2)-4 (a^3) (c^3)-(a^2) (c^4)-(b^2) (c^4)+2 a (c^5)+(c^6))(Sqrt[poly]));

d=(2 (((a-b))^2) (((a+b))^2) (a-b-c) (((b-c))^2) (a+b-c) (((-a+c))^2) (((a+c))^2) (a-b+c) (((b+c))^2) (a+b+c) )(2poly2-2 ((a^2)-(b^2)+a c+(c^2)) ((a^6)-(a^4) (b^2)+(a^2) (b^4)-(b^6)+2 (a^5) c-(a^4) (c^2)+(a^2) (b^2) (c^2)+(b^4) (c^2)-4 (a^3) (c^3)-(a^2) (c^4)-(b^2) (c^4)+2 a (c^5)+(c^6))(Sqrt[poly]));

ll=Factor[Expand[(((-((a^2)-(b^2)+a c+(c^2)) ((a^6)-(a^4) (b^2)+(a^2) (b^4)-(b^6)+2 (a^5) c-(a^4) (c^2)+(a^2) (b^2) (c^2)+(b^4) (c^2)-4 (a^3) (c^3)-(a^2) (c^4)-(b^2) (c^4)+2 a (c^5)+(c^6))+poly))^2)]/.poly^2:>((a^16)-4 (a^14) (b^2)+8 (a^12) (b^4)-12 (a^10) (b^6)+14 (a^8) (b^8)-12 (a^6) (b^10)+8 (a^4) (b^12)-4 (a^2) (b^14)+(b^16)+6 (a^15) c-26 (a^13) (b^2) c+50 (a^11) (b^4) c-62 (a^9) (b^6) c+58 (a^7) (b^8) c-38 (a^5) (b^10) c+14 (a^3) (b^12) c-2 a (b^14) c+13 (a^14) (c^2)-56 (a^12) (b^2) (c^2)+101 (a^10) (b^4) (c^2)-120 (a^8) (b^6) (c^2)+115 (a^6) (b^8) (c^2)-76 (a^4) (b^10) (c^2)+27 (a^2) (b^12) (c^2)-4 (b^14) (c^2)+14 (a^13) (c^3)-48 (a^11) (b^2) (c^3)+86 (a^9) (b^4) (c^3)-120 (a^7) (b^6) (c^3)+134 (a^5) (b^8) (c^3)-80 (a^3) (b^10) (c^3)+14 a (b^12) (c^3)+6 (a^12) (c^4)-2 (a^10) (b^2) (c^4)+31 (a^8) (b^4) (c^4)-104 (a^6) (b^6) (c^4)+141 (a^4) (b^8) (c^4)-76 (a^2) (b^10) (c^4)+8 (b^12) (c^4)-14 (a^11) (c^5)+30 (a^9) (b^2) (c^5)+2 (a^7) (b^4) (c^5)-102 (a^5) (b^6) (c^5)+134 (a^3) (b^8) (c^5)-38 a (b^10) (c^5)-29 (a^10) (c^6)+30 (a^8) (b^2) (c^6)-3 (a^6) (b^4) (c^6)-104 (a^4) (b^6) (c^6)+115 (a^2) (b^8) (c^6)-12 (b^10) (c^6)-6 (a^9) (c^7)+24 (a^7) (b^2) (c^7)+2 (a^5) (b^4) (c^7)-120 (a^3) (b^6) (c^7)+58 a (b^8) (c^7)+18 (a^8) (c^8)+30 (a^6) (b^2) (c^8)+31 (a^4) (b^4) (c^8)-120 (a^2) (b^6) (c^8)+14 (b^8) (c^8)-6 (a^7) (c^9)+30 (a^5) (b^2) (c^9)+86 (a^3) (b^4) (c^9)-62 a (b^6) (c^9)-29 (a^6) (c^10)-2 (a^4) (b^2) (c^10)+101 (a^2) (b^4) (c^10)-12 (b^6) (c^10)-14 (a^5) (c^11)-48 (a^3) (b^2) (c^11)+50 a (b^4) (c^11)+6 (a^4) (c^12)-56 (a^2) (b^2) (c^12)+8 (b^4) (c^12)+14 (a^3) (c^13)-26 a (b^2) (c^13)+13 (a^2) (c^14)-4 (b^2) (c^14)+6 a (c^15)+(c^16))]

f=Factor[Coefficient[ll,poly]]

Factor[ll-f poly]

2poly2-2 ((a^2)-(b^2)+a c+(c^2)) ((a^6)-(a^4) (b^2)+(a^2) (b^4)-(b^6)+2 (a^5) c-(a^4) (c^2)+(a^2) (b^2) (c^2)+(b^4) (c^2)-4 (a^3) (c^3)-(a^2) (c^4)-(b^2) (c^4)+2 a (c^5)+(c^6))poly

FullSimplify[Sqrt[Factor[Cancel[n/d]]],TriangleQ[]]

Map[Factor,%,3]

SecondSteinerCircle[t]

ToSidelengths[(1/(2a b c)) Circumradius√-((((a^6)-3 (a^4) (b^2)+3 (a^2) (b^4)-(b^6)-3 (a^4) (c^2)+3 (a^2) (b^2) (c^2)-(b^4) (c^2)+3 (a^2) (c^4)-(b^2) (c^4)-(c^6)) ((a^6)+(a^4) (b^2)+(a^2) (b^4)+(b^6)-3 (a^4) (c^2)-3 (a^2) (b^2) (c^2)-3 (b^4) (c^2)+3 (a^2) (c^4)+3 (b^2) (c^4)-(c^6)) ((a^6)-3 (a^4) (b^2)+3 (a^2) (b^4)-(b^6)+(a^4) (c^2)-3 (a^2) (b^2) (c^2)+3 (b^4) (c^2)+(a^2) (c^4)-3 (b^2) (c^4)+(c^6))))/Abs[(a^2-b^2)(a^2-c^2)(b^2-c^2)]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

InputForm[(1/(2a b c)) Circumradius√-((((a^6)-3 (a^4) (b^2)+3 (a^2) (b^4)-(b^6)-3 (a^4) (c^2)+3 (a^2) (b^2) (c^2)-(b^4) (c^2)+3 (a^2) (c^4)-(b^2) (c^4)-(c^6)) ((a^6)+(a^4) (b^2)+(a^2) (b^4)+(b^6)-3 (a^4) (c^2)-3 (a^2) (b^2) (c^2)-3 (b^4) (c^2)+3 (a^2) (c^4)+3 (b^2) (c^4)-(c^6)) ((a^6)-3 (a^4) (b^2)+3 (a^2) (b^4)-(b^6)+(a^4) (c^2)-3 (a^2) (b^2) (c^2)+3 (b^4) (c^2)+(a^2) (c^4)-3 (b^2) (c^4)+(c^6))))/Abs[(a^2-b^2)(a^2-c^2)(b^2-c^2)]]

eqn=Symmetrize[Numerator[Together//@ToSidelengths[TrilinearEquation[circ]]/.Sqrt[x_]Sqrt[y_]:>Simplify[Sqrt[x y],TriangleQ[]]]]

eqn=Symmetrize[Numerator[Together//@TrilinearEquation@@TrilinearVertexMatrix[SteinerTriangle]]]

lmn=Factor[CircleFunction[eqn]]

InputForm[lmn[[1]]]

IncidentCenters[KimberlingCenters,lmn]//Timing

t=Triangle[{{0,0},{1,0},{.0,.7}}];
Show[Graphics[{
t,
{Red,SecondSteinerCircle[t]},
{Blue,SteinerTriangle[t]}
}],AspectRatio->Automatic]