Kimberling Center

Author

Eric W. Weisstein
January 16, 2005

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

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

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

Initialization

Kimberling Lookup

KimberlingLookup[{1,1,1}]

KimberlingLookup[CyclicTrilinears[Cos[A]]]

KimberlingLookup[CyclicTrilinears[a+b]]

Position[KimberlingCoordinates,_?(Abs[ExactTrilinears[Triangle[6,9,13],{1,1,1}][[1]]-#]<10^-4&),1,1]

Position[KimberlingCoordinates,_?(Abs[ExactTrilinears[Triangle[6,9,13],CyclicTrilinears[Cos[A]]][[1]]-#]<10^-4&),1,1]

Brisse Centers

Length[brisse=#[[1,1,1]]&/@Select[DownValues[CenterFunction],#[[1,1,1,0]]==Brisse&&!StringQ[#[[2]]]&]]

t=Triangle[Table[Random[Real,{0,1},10],{3},{2}]]

abc=SideLengths[Evaluate[t]]

Bisymmetry Check

bisym={};
Do[
Print[{i,
t=TimeConstrained[FullSimplify[BisymmetricQ[brisse[[i]],Signed->True],TriangleQ[]],60]//Timing;AppendTo[bisym,t[[2]]];
t
}],
{i,Length[brisse]}]

Position[bisym,_?(#=!=True&),Heads->False]

Compare Kimberling and Brisse

Off[Power::"infy",∞::"indet",TrilinearToCartesian::"orient"]

l={};
Do[
If[!StringQ[CenterFunction[X[n]]]&&Head[CenterFunction[Brisse[n]]]=!=CenterFunction,
Print[l0={n,IncidentCenters[{X[n]},Trilinears[Brisse[n]],MaxIterations->1]}
];
AppendTo[l,l0];
],{n,kmax}]

Differences

First/@Select[l,Last[#]==={}&]

Moses Centers

Original Notebook (Now in package)

NotebookOpen["/Volumes/Users/eww/Mathematica/KimberlingCenters/Moses/ETCPoints.nb"];

Length[moses=#[[1,1,1]]&/@DownValues[x]]

(CenterFunction[Moses[#]]=x[#]/a/.{s->Semiperimeter,Δ->Area,J->((Sqrt[(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)])/(a b c)),e->(Sqrt[(((a^4)+(b^4)-(b^2) (c^2)+(c^4)-(a^2) ((b^2)+(c^2)))/((b^2) (c^2)+(a^2) ((b^2)+(c^2))))]),τ->GoldenRatio})&/@moses;

Bisymmetry Check

bisym={};
Do[
Print[{i,
t=TimeConstrained[FullSimplify[BisymmetricQ[CenterFunction[Moses[i]]/.TriangleRules,Signed->True],TriangleQ[]],60]//Timing;AppendTo[bisym,t[[2]]];
t
}],
{i,kmax}]

Position[bisym,_?(#=!=True&),Heads->False]

Simplify[Together//@ToSidelengths[BisymmetricQ[CenterFunction[Moses[2100]],Signed->True]],TriangleQ[]]//Timing

Factor//@(Together//@ToSidelengths[BisymmetricQ[CenterFunction[Moses[2101]],Signed->True]]/.sqrt_Sqrt:>Refine[sqrt,TriangleQ[]])

Compare KimberlingCenters.m and MosesCenters.m

Off[Power::"infy",∞::"indet",TrilinearToCartesian::"orient"]

l={};
Do[
If[!StringQ[CenterFunction[X[n]]]&&Head[CenterFunction[Moses[n]]]=!=CenterFunction,
Print[l0={n,IncidentCenters[{X[n]},CyclicTrilinears[CenterFunction[Moses[n]]],MaxIterations->1]}
];
AppendTo[l,l0];
],{n,kmax}]

Differences

difs=First/@Select[l,Last[#]==={}&]

Kimberling Centers

Count[Centers,_?(MatchQ[#,_X]&)]

kmax=Select[Centers,(MatchQ[#,_X]&)][[-1,1]]

Length[Select[Centers,(MatchQ[#,_X]&&!StringQ[CenterFunction[#]])&]]

(kimberling=KimberlingCenters)//Length

kimberling[[369]]

No Symbolic Triangle Center

Length[p=Position[CenterFunction[X[#]]&/@Range[kmax],_String,{1}]//Flatten]

{#,CenterName[X[#]],CenterFunction[X[#]]}&/@p//ColumnForm

Literal Duplications

Select[Split[Sort[{CenterFunction[#],#}&/@KimberlingCenters],First[#1]===First[#2]&],Length[#]>1&]//Timing

Numerical Duplications

t=Triangle[Table[Random[Real,{0,1},50],{3},{2}]];
kc=KimberlingCenters;
rules=Join[Thread[{a,b,c}->SideLengths[Evaluate[t]]],Thread[{A,B,C}->Angles[Evaluate[t]]]];

Length[k=Rest[#]/First[#]&/@((Trilinears/@KimberlingCenters)/.TriangleRules/.rules)]//Timing

n=Select[Split[Sort[k],#1==#2&],Length[#]>1&]

Sort[Map[kc[[#]]&,Function[pt,Flatten[Position[k,_?(#==pt&)]]]/@First/@n,{2}],(#1[[1,1]]<#2[[1,1]]&)]

272/1751

CenterName/@{X[272],X[1751]}

Through[{X[#],Brisse[#]}[t]]&/@{272,1751}

-Symmetrize[CenterFunction[Brisse[272]]]//InputForm

Symmetrize[Together//@(ToSidelengths[CenterFunction[X[272]]/.f:_Sin:>TrigExpand[f]]/.Sqrt[x_]:>FullSimplify[Sqrt[Expand[x]],TriangleQ[]])]//InputForm

Moses

(1/((b+c) ((a^2) b-(b^3)+(a^2) c+a b c-(c^3))a))

908/1532

CenterName/@{X[908],X[1532]}

Through[{X[#],Brisse[#]}[t]]&/@{908,1532}

Bisymmetry Check

{BisymmetricQ[(1/(b-c))],BisymmetricQ[(1/(b-c)),Signed->True]}

Simplify[%]

k=KimberlingCenters;
l={};
Do[
Print[{i,
t=TimeConstrained[FullSimplify[BisymmetricQ[k[[i]],Signed->True],TriangleQ[]],60]//Timing;AppendTo[l,t[[2]]];
t
}],
{i,Length[KimberlingCenters]}]

bad=KimberlingCenters[[Position[l,_?(#=!=True&),{1},Heads->False]//Flatten]]

{#,CenterName[#]}&/@bad//ColumnForm

By Hand

Discrepancies

(Subscript[X, 166])

Cartesian[Intersections[Line[X[165],X[168]],Line[X[167],X[188]]][t]]

X[166][t]

Brisse[166][t]

Together//@ToSidelengths[HalfAngleFormulas[Intersections[Line[X[165],X[168]],Line[X[167],X[188]]]/.f:(_Sin|_Cos):>TrigExpand[f]]]/.Sqrt[x_]Sqrt[y_]:>FullSimplify[Sqrt[ExpandAll[x y]],TriangleQ[]]/. 1/(Sqrt[x_]Sqrt[y_]):>FullSimplify[1/Sqrt[ExpandAll[x y]],TriangleQ[]]

Cartesian[CyclicTrilinears[(Tan[ A/2])/(Cos [B/2]+Cos[ C/2]-Cos[ A/2])-(Tan [B/2])/(Cos[ C/2]+Cos[ A/2]-Cos[ B/2])-(Tan [C/2])/(Cos[ A/2]+Cos[ B/2]-Cos[ C/2])][t]]

(Subscript[X, 167])

s[a_,b_,c_]:=Sin[a/2]
t[a_,b_,c_]:=Cos[b/2]+Cos[c/2]-Cos[a/2]

x167=InputForm[s[B,C,A]t[B,C,A]+s[C,A,B]t[C,A,B]-s[A,B,C]t[A,B,C]]

(DistanceTrilinears[Together//@HalfAngleFormulas[x167]]/.Sqrt[x_]Sqrt[y_]:>Simplify[Sqrt[Together//@(x y)],TriangleQ[]]/.(2/(Sqrt[((x_)/(y_))])):>2Simplify[(Sqrt[(y/x)]),TriangleQ[]])/.Sqrt[x_]:>FullSimplify[Sqrt[x],TriangleQ[]]

{Brisse[167][t],X[167][t]}

CenterFunction[Brisse[167]]

CenterFunction[Brisse[168]]

{Brisse[168][t],X[168][t]}

(Subscript[X, 180])

ETC is wrong

f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/t(B,C,A) + 1/t(C,B,A) - 1/t(C,A,B),
                        t(A,B,C) = 1 + 2(sec A/4 cos B/4 cos C/4)2

t[A_,B_,C_]:=1+2(Sec[A/4]Cos[B/4]Cos[C/4])^2

Plus@@(1/{t[B,C,A],t[C,B,A],-t[C,A,B]})//InputForm

(Subscript[X, 260])

CenterFunction[X[260]]

CenterFunction[Brisse[260]]

{X[260][t],Brisse[260][t],Cartesian[CyclicTrilinears[Cos[A/2]/(Cos[B/2]+Cos[C/2])][t]]}

(Subscript[X, 272])

CenterName/@{X[272],X[1751]}

Through[{X[#],Brisse[#]}[t]]&/@{272,1751}

-Symmetrize[CenterFunction[Brisse[272]]]//InputForm

Symmetrize[Together//@(ToSidelengths[CenterFunction[X[272]]/.f:_Sin:>TrigExpand[f]]/.Sqrt[x_]:>FullSimplify[Sqrt[Expand[x]],TriangleQ[]])]//InputForm

Moses

(1/((b+c) ((a^2) b-(b^3)+(a^2) c+a b c-(c^3))a))

{462,"Euler line intercept of line X(51)X(397)","u Tan[A]"}

eqn=Together//@DistanceTrilinears[TrigExpand[Intersections[Trilinears/@X/@{4,5},Trilinears/@X/@{51,397}]][[1]]]

Together//@(eqn/.Sqrt[x_]:>FullSimplify[Sqrt[x],TriangleQ[]])

Symmetrize[%]

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

(1/(a ((a^2)-(b^2)-(c^2))))((a-b-c) (a+b-c) (a-b+c) (a+b+c)+(Sqrt[3]) (a^2) (Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)]))

FullSimplify[Together//@DistanceTrilinears[Tan[A]TrigExpand[CenterFunction[X[395]]]],TriangleQ[]]

Symmetrize[%]

Tan[A]CenterFunction[X[395]]//InputForm

{463,"Euler line intercept of line X(51)X(398)","u Tan[A]"}

Tan[A]CenterFunction[X[396]]//InputForm

{465,"Euler line intercept of line X(216)X(395)","u Cot[A]"}

Cot[A]CenterFunction[X[397]]//InputForm

{466,"Euler line intercept of line X(216)X(396)","u Cot[A]"}

Cot[A]CenterFunction[X[398]]//InputForm

(Subscript[X, 369])

TriangleQ[]

t=.

Reduce[2t^3-3(a+b+c)t^2+(a^2+b^2+c^2+8b c+8c a+8a b)t-(c b^2+a c^2+b a^2+5b c^2+5c a^2+5a b^2+9a b c)==0&&a>0&&b>0&&c>0&&a+b>c&&b+c>a&&a+c>b,t,Reals]

Function[r,b c(r^2-(2c+a)r+(-a^2+b^2+2c^2+2b c+3c a+2a b))][Root[-(a^2) b-5 a (b^2)-5 (a^2) c-9 a b c-(b^2) c-a (c^2)-5 b (c^2)+(a^2) #1+8 a b #1+(b^2) #1+8 a c #1+8 b c #1+(c^2) #1-3 a ((#1)^2)-3 b ((#1)^2)-3 c ((#1)^2)+2 ((#1)^3)&,1]]

(Subscript[X, 681])

CenterName[X[681]]

Symmetrize[CenterFunction[Brisse[681]]]

%//InputForm

Symmetrize[Factor[ToSidelengths[CenterFunction[X[681]]]/.Sqrt[x_]:>FullSimplify[Sqrt[Together[x]],TriangleQ[]]]]

InputForm[%]

Symmetrize[Factor[Together//@DistanceTrilinears[TrilinearPole[LineEquation[{X[6],X[158]}]][[1]]]]]//Factor

Symmetrize[Together//@ToSidelengths[Times@@(CenterFunction/@{Brisse[681],X[681]})]/.Sqrt[x_]:>FullSimplify[Sqrt[x],TriangleQ[]]]

(Subscript[X, 741])

CenterName[X[741]]

(Subscript[X, 743])

CenterName[X[743]]

(Subscript[X, 745])

CenterName[X[745]]

(Subscript[X, 747])

CenterName[X[747]]

(Subscript[X, 753])

CenterName[X[753]]

(Subscript[X, 755])

CenterName[X[755]]

(Subscript[X, 759])

CenterName[X[759]]

(Subscript[X, 761])

Symmetrize[CenterFunction[X[761]]]

Symmetrize[CenterFunction[Brisse[761]]]

(Subscript[X, 767])

CenterName[X[767]]

Symmetrize[CenterFunction[X[767]]]

Symmetrize[CenterFunction[Brisse[767]]]

(Subscript[X, 844])

x=Sin[A/2]Sec[A/2]^2;
y=Sin[B/2]Sec[B/2]^2;
z=Sin[C/2]Sec[C/2]^2;

{844,"intersection of lines X(166)X(167) and X(173)X(503)","-x+y+z, x = x(A,B,C)= Sin[A/2]Sec[A/2]^2"}

InputForm[-x+y+z]

Intersections[Trilinears/@X/@{166,167},Trilinears/@X/@{173,503}]

(Subscript[X, 845])

{845,"intersection of lines X(165)X(166) and X(173)X(503)","-x+y+z, x = x(A,B,C)= Sin[A/2]^2Sec[A/2]"}

x=Sin[A/2]^2Sec[A/2];
y=Sin[B/2]^2Sec[B/2];
z=Sin[C/2]^2Sec[C/2];

InputForm[-x+y+z]

Simplify[Together//@DistanceTrilinears[HalfAngleFormulas[-x+y+z]],TriangleQ[]]

Intersections[X/@{165,167},X/@{173,503}]

(Subscript[X, 848])

CenterFunction[X[848]]

CenterFunction[Brisse[848]]

(Subscript[X, 1012])

CenterFunction[X[1012]]

CenterFunction[Brisse[1012]]//InputForm

(Subscript[X, 1113])

CircleEquation[Circumcircle]

soln=Solve[{LineEquation[EulerLine]==0,CircleEquation[Circumcircle]==0,2Area[]==(a α+b β+c γ)},{α,β,γ}]

soln2=Together//@DistanceTrilinears[Last/@Sort/@soln[[1]]]

Factor[ z1=((a^22)-9 (a^20) (b^2)+35 (a^18) (b^4)-75 (a^16) (b^6)+90 (a^14) (b^8)-42 (a^12) (b^10)-42 (a^10) (b^12)+90 (a^8) (b^14)-75 (a^6) (b^16)+35 (a^4) (b^18)-9 (a^2) (b^20)+(b^22)-6 (a^20) (c^2)+41 (a^18) (b^2) (c^2)-118 (a^16) (b^4) (c^2)+188 (a^14) (b^6) (c^2)-196 (a^12) (b^8) (c^2)+182 (a^10) (b^10) (c^2)-196 (a^8) (b^12) (c^2)+188 (a^6) (b^14) (c^2)-118 (a^4) (b^16) (c^2)+41 (a^2) (b^18) (c^2)-6 (b^20) (c^2)+11 (a^18) (c^4)-58 (a^16) (b^2) (c^4)+125 (a^14) (b^4) (c^4)-137 (a^12) (b^6) (c^4)+59 (a^10) (b^8) (c^4)+59 (a^8) (b^10) (c^4)-137 (a^6) (b^12) (c^4)+125 (a^4) (b^14) (c^4)-58 (a^2) (b^16) (c^4)+11 (b^18) (c^4)+6 (a^16) (c^6)-(a^14) (b^2) (c^6)-26 (a^12) (b^4) (c^6)+17 (a^10) (b^6) (c^6)+8 (a^8) (b^8) (c^6)+17 (a^6) (b^10) (c^6)-26 (a^4) (b^12) (c^6)-(a^2) (b^14) (c^6)+6 (b^16) (c^6)-54 (a^14) (c^8)+77 (a^12) (b^2) (c^8)-19 (a^10) (b^4) (c^8)-4 (a^8) (b^6) (c^8)-4 (a^6) (b^8) (c^8)-19 (a^4) (b^10) (c^8)+77 (a^2) (b^12) (c^8)-54 (b^14) (c^8)+84 (a^12) (c^10)-91 (a^10) (b^2) (c^10)-40 (a^8) (b^4) (c^10)-34 (a^6) (b^6) (c^10)-40 (a^4) (b^8) (c^10)-91 (a^2) (b^10) (c^10)+84 (b^12) (c^10)-42 (a^10) (c^12)+119 (a^8) (b^2) (c^12)+163 (a^6) (b^4) (c^12)+163 (a^4) (b^6) (c^12)+119 (a^2) (b^8) (c^12)-42 (b^10) (c^12)-36 (a^8) (c^14)-187 (a^6) (b^2) (c^14)-250 (a^4) (b^4) (c^14)-187 (a^2) (b^6) (c^14)-36 (b^8) (c^14)+69 (a^6) (c^16)+176 (a^4) (b^2) (c^16)+176 (a^2) (b^4) (c^16)+69 (b^6) (c^16)-46 (a^4) (c^18)-82 (a^2) (b^2) (c^18)-46 (b^4) (c^18)+15 (a^2) (c^20)+15 (b^2) (c^20)-2 (c^22))]

Factor[z2=(-(a^34)+15 (a^32) (b^2)-104 (a^30) (b^4)+440 (a^28) (b^6)-1260 (a^26) (b^8)+2548 (a^24) (b^10)-3640 (a^22) (b^12)+3432 (a^20) (b^14)-1430 (a^18) (b^16)-1430 (a^16) (b^18)+3432 (a^14) (b^20)-3640 (a^12) (b^22)+2548 (a^10) (b^24)-1260 (a^8) (b^26)+440 (a^6) (b^28)-104 (a^4) (b^30)+15 (a^2) (b^32)-(b^34)+9 (a^32) (c^2)-113 (a^30) (b^2) (c^2)+646 (a^28) (b^4) (c^2)-2219 (a^26) (b^6) (c^2)+5096 (a^24) (b^8) (c^2)-8281 (a^22) (b^10) (c^2)+10010 (a^20) (b^12) (c^2)-9867 (a^18) (b^14) (c^2)+9438 (a^16) (b^16) (c^2)-9867 (a^14) (b^18) (c^2)+10010 (a^12) (b^20) (c^2)-8281 (a^10) (b^22) (c^2)+5096 (a^8) (b^24) (c^2)-2219 (a^6) (b^26) (c^2)+646 (a^4) (b^28) (c^2)-113 (a^2) (b^30) (c^2)+9 (b^32) (c^2)-29 (a^30) (c^4)+301 (a^28) (b^2) (c^4)-1397 (a^26) (b^4) (c^4)+3813 (a^24) (b^6) (c^4)-6769 (a^22) (b^8) (c^4)+8129 (a^20) (b^10) (c^4)-6457 (a^18) (b^12) (c^4)+2409 (a^16) (b^14) (c^4)+2409 (a^14) (b^16) (c^4)-6457 (a^12) (b^18) (c^4)+8129 (a^10) (b^20) (c^4)-6769 (a^8) (b^22) (c^4)+3813 (a^6) (b^24) (c^4)-1397 (a^4) (b^26) (c^4)+301 (a^2) (b^28) (c^4)-29 (b^30) (c^4)+17 (a^28) (c^6)-152 (a^26) (b^2) (c^6)+591 (a^24) (b^4) (c^6)-1272 (a^22) (b^6) (c^6)+1517 (a^20) (b^8) (c^6)-584 (a^18) (b^10) (c^6)-1101 (a^16) (b^12) (c^6)+1968 (a^14) (b^14) (c^6)-1101 (a^12) (b^16) (c^6)-584 (a^10) (b^18) (c^6)+1517 (a^8) (b^20) (c^6)-1272 (a^6) (b^22) (c^6)+591 (a^4) (b^24) (c^6)-152 (a^2) (b^26) (c^6)+17 (b^28) (c^6)+147 (a^26) (c^8)-949 (a^24) (b^2) (c^8)+2606 (a^22) (b^4) (c^8)-4018 (a^20) (b^6) (c^8)+3925 (a^18) (b^8) (c^8)-2579 (a^16) (b^10) (c^8)+868 (a^14) (b^12) (c^8)+868 (a^12) (b^14) (c^8)-2579 (a^10) (b^16) (c^8)+3925 (a^8) (b^18) (c^8)-4018 (a^6) (b^20) (c^8)+2606 (a^4) (b^22) (c^8)-949 (a^2) (b^24) (c^8)+147 (b^26) (c^8)-455 (a^24) (c^10)+2405 (a^22) (b^2) (c^10)-5164 (a^20) (b^4) (c^10)+5665 (a^18) (b^6) (c^10)-3185 (a^16) (b^8) (c^10)+634 (a^14) (b^10) (c^10)+200 (a^12) (b^12) (c^10)+634 (a^10) (b^14) (c^10)-3185 (a^8) (b^16) (c^10)+5665 (a^6) (b^18) (c^10)-5164 (a^4) (b^20) (c^10)+2405 (a^2) (b^22) (c^10)-455 (b^24) (c^10)+455 (a^22) (c^12)-2431 (a^20) (b^2) (c^12)+4373 (a^18) (b^4) (c^12)-3117 (a^16) (b^6) (c^12)+598 (a^14) (b^8) (c^12)+122 (a^12) (b^10) (c^12)+122 (a^10) (b^12) (c^12)+598 (a^8) (b^14) (c^12)-3117 (a^6) (b^16) (c^12)+4373 (a^4) (b^18) (c^12)-2431 (a^2) (b^20) (c^12)+455 (b^22) (c^12)+429 (a^20) (c^14)+858 (a^18) (b^2) (c^14)-2271 (a^16) (b^4) (c^14)+744 (a^14) (b^6) (c^14)+178 (a^12) (b^8) (c^14)+124 (a^10) (b^10) (c^14)+178 (a^8) (b^12) (c^14)+744 (a^6) (b^14) (c^14)-2271 (a^4) (b^16) (c^14)+858 (a^2) (b^18) (c^14)+429 (b^20) (c^14)-1859 (a^18) (c^16)+429 (a^16) (b^2) (c^16)+1788 (a^14) (b^4) (c^16)+924 (a^12) (b^6) (c^16)+766 (a^10) (b^8) (c^16)+766 (a^8) (b^10) (c^16)+924 (a^6) (b^12) (c^16)+1788 (a^4) (b^14) (c^16)+429 (a^2) (b^16) (c^16)-1859 (b^18) (c^16)+2431 (a^16) (c^18)-1287 (a^14) (b^2) (c^18)-4042 (a^12) (b^4) (c^18)-5129 (a^10) (b^6) (c^18)-5450 (a^8) (b^8) (c^18)-5129 (a^6) (b^10) (c^18)-4042 (a^4) (b^12) (c^18)-1287 (a^2) (b^14) (c^18)+2431 (b^16) (c^18)-1287 (a^14) (c^20)+3575 (a^12) (b^2) (c^20)+9125 (a^10) (b^4) (c^20)+12395 (a^8) (b^6) (c^20)+12395 (a^6) (b^8) (c^20)+9125 (a^4) (b^10) (c^20)+3575 (a^2) (b^12) (c^20)-1287 (b^14) (c^20)-637 (a^12) (c^22)-6604 (a^10) (b^2) (c^22)-13519 (a^8) (b^4) (c^22)-16464 (a^6) (b^6) (c^22)-13519 (a^4) (b^8) (c^22)-6604 (a^2) (b^10) (c^22)-637 (b^12) (c^22)+1729 (a^10) (c^24)+7241 (a^8) (b^2) (c^24)+12318 (a^6) (b^4) (c^24)+12318 (a^4) (b^6) (c^24)+7241 (a^2) (b^8) (c^24)+1729 (b^10) (c^24)-1533 (a^8) (c^26)-4877 (a^6) (b^2) (c^26)-6704 (a^4) (b^4) (c^26)-4877 (a^2) (b^6) (c^26)-1533 (b^8) (c^26)+797 (a^6) (c^28)+2011 (a^4) (b^2) (c^28)+2011 (a^2) (b^4) (c^28)+797 (b^6) (c^28)-257 (a^4) (c^30)-470 (a^2) (b^2) (c^30)-257 (b^4) (c^30)+48 (a^2) (c^32)+48 (b^2) (c^32)-4 (c^34))]

Together//@((Together//@Refine[soln2/.{z1->Factor[z1],z2->Factor[z2]},TriangleQ[]])/.(Sqrt[(a+b-c) (a-b+c) (a+b+c)]) (Sqrt[-(((a-b-c))^5) xx_]):>(((a-b-c))^2)(Sqrt[-(a-b-c)(a+b-c) (a-b+c) (a+b+c)])(Sqrt[xx])/.Abs[x_]:>x)

Factor[%]

Symmetrize/@%%

Factor[Coefficient[-2 (a^4) b c √((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))+(a^2) (b^3) c √((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))+(b^5) c √((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))+(a^2) b (c^3) √((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))-2 (b^3) (c^3) √((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))+b (c^5) √((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)),√((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))]]

Factor[(a^9)-2 (a^7) (b^2)+2 (a^3) (b^6)-a (b^8)-2 (a^7) (c^2)+5 (a^5) (b^2) (c^2)-3 (a^3) (b^4) (c^2)-3 (a^3) (b^2) (c^4)+2 a (b^4) (c^4)+2 (a^3) (c^6)-a (c^8)]

a ((a^2)-(b^2)-(c^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))-b c (2 (a^4)-(a^2) (b^2)-(b^4)-(a^2) (c^2)+2 (b^2) (c^2)-(c^4)) √((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))

a ((a^2)-(b^2)-(c^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))-b c (2 (a^4)-(a^2) (b^2)-(b^4)-(a^2) (c^2)+2 (b^2) (c^2)-(c^4))

Together//@DistanceTrilinears[(1-(1/(a b c))√((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)))Cos[A]-2 Cos[B]Cos[C]]

(-Factor[2 (a^4) b c+(a^2) (b^3) c+(b^5) c+(a^2) b (c^3)-2 (b^3) (c^3)+b (c^5)]+Factor[(a^3) -a (b^2)-a (c^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)))

(Subscript[X, 1344])

CenterFunction[X[1344]]

Distance[Circumcenter,Orthocenter]

Factor[Symmetrize[ToSidelengths[CenterFunction[X[1344]]]]]

Factor[-CenterFunction[Brisse[1344]]]

Symmetrize[ToSidelengths[(a^2) (1+J) SA+4 SB SC/.J->((Sqrt[(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)])/(a b c))]/a]

(Subscript[X, 1376])

CenterName[X[1376]]

CenterFunction[X[1376]]//InputForm

Symmetrize[CenterFunction[X[1376]]]//InputForm

Symmetrize[CenterFunction[Brisse[1376]]]//InputForm

Symmetrize[Simplify[Together//@DistanceTrilinears[ExternalSimilitudeCenter[Circumcircle,SpiekerCircle][[1]]],TriangleQ[]]]

(Subscript[X, 1453])

CenterName[X[1453]]

Symmetrize[ToSidelengths[BethConjugate[X[452],X[452]][[1]]]]

CenterFunction[Brisse[1453]]

CenterFunction[X[1453]]

(Subscript[X, 1511])

CenterFunction/@{X[526],X[1511]}

CenterName/@{X[526],X[1511]}

Symmetrize[Simplify[Together//@DistanceTrilinears[TrigExpand[CenterFunction[X[526]]]],TriangleQ[]]]

Symmetrize[Simplify[Together//@DistanceTrilinears[TrigExpand[CenterFunction[X[1511]]]],TriangleQ[]]]

Symmetrize[Simplify[Together//@DistanceTrilinears[TrigExpand[CyclicTrilinears[(Sin[A]^2-3Cos[A]^2)(Sin[B]Sin[C]-3Cos[B]Cos[C])]]],TriangleQ[]][[1]]]

CenterFunction[Brisse[#]]&/@{526,1511}

(Subscript[X, 1532])

CenterName[X[1532]]

TrilinearPolar[{l_,m_,n_}]:=m n α+n l β+l m γ
TrilinearPolar[ctr_]:=Trilinears[ctr]/;MemberQ[Centers,ctr]

Orthopole[line_]:=Orthopole[Trilinears[line]]/;MemberQ[Lines,line]
Orthopole[eqn_]:=Orthopole[Coefficient[eqn,#]&/@{α,β,γ}]
Null

TrilinearPolar[IsogonalConjugate[X[37]]]

Factor[((2a b c(((a+b))^2) (((a+c))^2) (((b+c))^2))/((a+b+c) ))Together//@DistanceTrilinears[Orthopole[%143][[1]]]]

CenterFunction[X[1532]]//InputForm

CenterFunction[Brisse[1532]]//InputForm

Factor[((2 a b (((a+b))^2) c (((a+c))^2) (((b+c))^2))/(((a+b+c))^2))DistanceTrilinears[Orthojoin[X[37]]][[1]]//Simplify]

InputForm[%59]

X[37]

Orthopole[Trilinears[X[37]]]

Factor[((2a b c)/((((a+b+c))^2) ))DistanceTrilinears[Orthopole[Trilinears[X[37]]]][[1]]//Simplify]

(Subscript[X, 1563])

Symmetrize[Together//@DistanceTrilinears[Orthojoin[X[588]][[1]]]]

Together//@ExpandAll[Together//@DistanceTrilinears[Orthojoin[X[588]][[1]]]]

eqn=Symmetrize[%]

(1/a)({1,(Sqrt[Factor[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)]])}.(Factor/@(Coefficient[a eqn,(Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)]),#]&/@{0,1})))

InputForm[%18]

(Subscript[X, 1564])

eqn=Symmetrize[Together//@ExpandAll[Together//@DistanceTrilinears[Orthojoin[X[589]][[1]]]]]

(1/a)({1,(Sqrt[Factor[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)]])}.(Factor/@(Coefficient[a eqn,(Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)]),#]&/@{0,1})))

InputForm[%]

(Subscript[X, 1682])

CenterName[X[1682]]

Simplify[Together//@ExpandAll[Together//@(CenterFunction[X[1682]]//HalfAngleFormulas//ToSidelengths)],TriangleQ[]]

Symmetrize[%]

Symmetrize[Simplify[Together//@ToSidelengths[HalfAngleFormulas[CenterFunction[X[1682]]]],TriangleQ[]]]//InputForm

t

Cartesian[CyclicTrilinears[-a (a-b-c) (((a b+(b^2)+a c+(c^2)))^2)][t]]

CenterFunction[Brisse[1682]]//InputForm

Cartesian[CyclicTrilinears[- (a-b-c) (((a b+(b^2)+a c+(c^2)))^2)][t]]

rs=CircleRadius/@{Incircle,ApolloniusCircle}/.TriangleRules/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

ctrs=Center/@{Incircle,ApolloniusCircle}/.TriangleRules/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

Cartesian[(Plus@@(Reverse[rs] ctrs)/Plus@@rs)[t]]

{ctrs,rs}=Transpose[{Center[#][[1]],CircleRadius[#]}&/@{Circle[Incircle][t],Circle[ApolloniusCircle][t]}]

(Plus@@(Reverse[rs] ctrs)/Plus@@rs)

Symmetrize[Together//@ToSidelengths[InternalSimilitudeCenter[Incircle,ApolloniusCircle]][[1]]]

Symmetrize/@(Together//@InternalSimilitudeCenter[Incircle,ApolloniusCircle])

Circle/@{Incircle,ApolloniusCircle}//ToSidelengths

ExactTrilinears[Center[Incircle]]//ToSidelengths

Center[ApolloniusCircle]//ToSidelengths

Factor[CircleRadius[ApolloniusCircle]^2//ToSidelengths]

rules=Join[Thread[{α1,β1,γ1}->{1,1,1}],Thread[{α2,β2,γ2}->Center[ApolloniusCircle]],
{
r1->CircleRadius[Incircle]//ToSidelengths,
r2->Factor[CircleRadius[ApolloniusCircle]//ToSidelengths]
}]

eqns=Numerator/@Factor[{(((a α1+b β1+c γ1))^2) (((a α2+b β2+c γ2))^2)(d1+d2)^2+a b c (-a b c ((α2)^2) ((β1)^2)+2 a b c α1 α2 β1 β2-a b c ((α1)^2) ((β2)^2)+(a^3) ((α2)^2) β1 γ1-a (b^2) ((α2)^2) β1 γ1-a (c^2) ((α2)^2) β1 γ1-(a^3) α1 α2 β2 γ1+a (b^2) α1 α2 β2 γ1+a (c^2) α1 α2 β2 γ1+(a^2) b α2 β1 β2 γ1-(b^3) α2 β1 β2 γ1+b (c^2) α2 β1 β2 γ1-(a^2) b α1 ((β2)^2) γ1+(b^3) α1 ((β2)^2) γ1-b (c^2) α1 ((β2)^2) γ1-a b c ((α2)^2) ((γ1)^2)-(a^2) c α2 β2 ((γ1)^2)-(b^2) c α2 β2 ((γ1)^2)+(c^3) α2 β2 ((γ1)^2)-a b c ((β2)^2) ((γ1)^2)-(a^3) α1 α2 β1 γ2+a (b^2) α1 α2 β1 γ2+a (c^2) α1 α2 β1 γ2-(a^2) b α2 ((β1)^2) γ2+(b^3) α2 ((β1)^2) γ2-b (c^2) α2 ((β1)^2) γ2+(a^3) ((α1)^2) β2 γ2-a (b^2) ((α1)^2) β2 γ2-a (c^2) ((α1)^2) β2 γ2+(a^2) b α1 β1 β2 γ2-(b^3) α1 β1 β2 γ2+b (c^2) α1 β1 β2 γ2+2 a b c α1 α2 γ1 γ2+(a^2) c α2 β1 γ1 γ2+(b^2) c α2 β1 γ1 γ2-(c^3) α2 β1 γ1 γ2+(a^2) c α1 β2 γ1 γ2+(b^2) c α1 β2 γ1 γ2-(c^3) α1 β2 γ1 γ2+2 a b c β1 β2 γ1 γ2-a b c ((α1)^2) ((γ2)^2)-(a^2) c α1 β1 ((γ2)^2)-(b^2) c α1 β1 ((γ2)^2)+(c^3) α1 β1 ((γ2)^2)-a b c ((β1)^2) ((γ2)^2)),
(((a α+b β+c γ))^2) (((a α1+b β1+c γ1))^2)d1^2+a b c (-a b c ((α1)^2) (β^2)+2 a b c α α1 β β1-a b c (α^2) ((β1)^2)+(a^3) ((α1)^2) β γ-a (b^2) ((α1)^2) β γ-a (c^2) ((α1)^2) β γ-(a^3) α α1 β1 γ+a (b^2) α α1 β1 γ+a (c^2) α α1 β1 γ+(a^2) b α1 β β1 γ-(b^3) α1 β β1 γ+b (c^2) α1 β β1 γ-(a^2) b α ((β1)^2) γ+(b^3) α ((β1)^2) γ-b (c^2) α ((β1)^2) γ-a b c ((α1)^2) (γ^2)-(a^2) c α1 β1 (γ^2)-(b^2) c α1 β1 (γ^2)+(c^3) α1 β1 (γ^2)-a b c ((β1)^2) (γ^2)-(a^3) α α1 β γ1+a (b^2) α α1 β γ1+a (c^2) α α1 β γ1-(a^2) b α1 (β^2) γ1+(b^3) α1 (β^2) γ1-b (c^2) α1 (β^2) γ1+(a^3) (α^2) β1 γ1-a (b^2) (α^2) β1 γ1-a (c^2) (α^2) β1 γ1+(a^2) b α β β1 γ1-(b^3) α β β1 γ1+b (c^2) α β β1 γ1+2 a b c α α1 γ γ1+(a^2) c α1 β γ γ1+(b^2) c α1 β γ γ1-(c^3) α1 β γ γ1+(a^2) c α β1 γ γ1+(b^2) c α β1 γ γ1-(c^3) α β1 γ γ1+2 a b c β β1 γ γ1-a b c (α^2) ((γ1)^2)-(a^2) c α β ((γ1)^2)-(b^2) c α β ((γ1)^2)+(c^3) α β ((γ1)^2)-a b c (β^2) ((γ1)^2)),
(((a α+b β+c γ))^2) (((a α2+b β2+c γ2))^2)d2^2+a b c (-a b c ((α2)^2) (β^2)+2 a b c α α2 β β2-a b c (α^2) ((β2)^2)+(a^3) ((α2)^2) β γ-a (b^2) ((α2)^2) β γ-a (c^2) ((α2)^2) β γ-(a^3) α α2 β2 γ+a (b^2) α α2 β2 γ+a (c^2) α α2 β2 γ+(a^2) b α2 β β2 γ-(b^3) α2 β β2 γ+b (c^2) α2 β β2 γ-(a^2) b α ((β2)^2) γ+(b^3) α ((β2)^2) γ-b (c^2) α ((β2)^2) γ-a b c ((α2)^2) (γ^2)-(a^2) c α2 β2 (γ^2)-(b^2) c α2 β2 (γ^2)+(c^3) α2 β2 (γ^2)-a b c ((β2)^2) (γ^2)-(a^3) α α2 β γ2+a (b^2) α α2 β γ2+a (c^2) α α2 β γ2-(a^2) b α2 (β^2) γ2+(b^3) α2 (β^2) γ2-b (c^2) α2 (β^2) γ2+(a^3) (α^2) β2 γ2-a (b^2) (α^2) β2 γ2-a (c^2) (α^2) β2 γ2+(a^2) b α β β2 γ2-(b^3) α β β2 γ2+b (c^2) α β β2 γ2+2 a b c α α2 γ γ2+(a^2) c α2 β γ γ2+(b^2) c α2 β γ γ2-(c^3) α2 β γ γ2+(a^2) c α β2 γ γ2+(b^2) c α β2 γ γ2-(c^3) α β2 γ γ2+2 a b c β β2 γ γ2-a b c (α^2) ((γ2)^2)-(a^2) c α β ((γ2)^2)-(b^2) c α β ((γ2)^2)+(c^3) α β ((γ2)^2)-a b c (β^2) ((γ2)^2)),
(d1^2) (r2^2)-(d2^2) (r1^2),
4(a α+b β+c γ)^2-(a+b-c) (a-b+c) (-a+b+c) (a+b+c)
}/.rules]

eqns={((a^6)+4 (a^5) b-6 (a^3) (b^3)+4 a (b^5)+(b^6)+4 (a^5) c+10 (a^4) b c-8 (a^3) (b^2) c-8 (a^2) (b^3) c+10 a (b^4) c+4 (b^5) c-8 (a^3) b (c^2)+9 (a^2) (b^2) (c^2)-8 a (b^3) (c^2)-6 (a^3) (c^3)-8 (a^2) b (c^3)-8 a (b^2) (c^3)-6 (b^3) (c^3)+10 a b (c^4)+4 a (c^5)+4 b (c^5)+(c^6)+4 (a^4) (d1^2)-8 (a^2) (b^2) (d1^2)+4 (b^4) (d1^2)-8 (a^2) (c^2) (d1^2)-8 (b^2) (c^2) (d1^2)+4 (c^4) (d1^2)+8 (a^4) d1 d2-16 (a^2) (b^2) d1 d2+8 (b^4) d1 d2-16 (a^2) (c^2) d1 d2-16 (b^2) (c^2) d1 d2+8 (c^4) d1 d2+4 (a^4) (d2^2)-8 (a^2) (b^2) (d2^2)+4 (b^4) (d2^2)-8 (a^2) (c^2) (d2^2)-8 (b^2) (c^2) (d2^2)+4 (c^4) (d2^2)),
((a^3) b c (α^2)-(a^2) (b^2) c (α^2)-(a^2) b (c^2) (α^2)+(a^3) (d1^2) (α^2)+(a^2) b (d1^2) (α^2)+(a^2) c (d1^2) (α^2)-(a^3) b c α β+2 (a^2) (b^2) c α β-a (b^3) c α β+a b (c^3) α β+2 (a^2) b (d1^2) α β+2 a (b^2) (d1^2) α β+2 a b c (d1^2) α β-(a^2) (b^2) c (β^2)+a (b^3) c (β^2)-a (b^2) (c^2) (β^2)+a (b^2) (d1^2) (β^2)+(b^3) (d1^2) (β^2)+(b^2) c (d1^2) (β^2)-(a^3) b c α γ+a (b^3) c α γ+2 (a^2) b (c^2) α γ-a b (c^3) α γ+2 (a^2) c (d1^2) α γ+2 a b c (d1^2) α γ+2 a (c^2) (d1^2) α γ+(a^3) b c β γ-a (b^3) c β γ+2 a (b^2) (c^2) β γ-a b (c^3) β γ+2 a b c (d1^2) β γ+2 (b^2) c (d1^2) β γ+2 b (c^2) (d1^2) β γ-(a^2) b (c^2) (γ^2)-a (b^2) (c^2) (γ^2)+a b (c^3) (γ^2)+a (c^2) (d1^2) (γ^2)+b (c^2) (d1^2) (γ^2)+(c^3) (d1^2) (γ^2)),
((a^8) (α^2)+2 (a^7) b (α^2)-2 (a^5) (b^3) (α^2)+2 (a^3) (b^5) (α^2)+(a^2) (b^6) (α^2)+2 (a^7) c (α^2)+6 (a^6) b c (α^2)+2 (a^5) (b^2) c (α^2)-4 (a^4) (b^3) c (α^2)+4 (a^2) (b^5) c (α^2)+2 a (b^6) c (α^2)+2 (a^5) b (c^2) (α^2)+(a^4) (b^2) (c^2) (α^2)-4 (a^3) (b^3) (c^2) (α^2)+2 a (b^5) (c^2) (α^2)-2 (a^5) (c^3) (α^2)-4 (a^4) b (c^3) (α^2)-4 (a^3) (b^2) (c^3) (α^2)-6 (a^2) (b^3) (c^3) (α^2)-4 a (b^4) (c^3) (α^2)-4 a (b^3) (c^4) (α^2)+2 (a^3) (c^5) (α^2)+4 (a^2) b (c^5) (α^2)+2 a (b^2) (c^5) (α^2)+(a^2) (c^6) (α^2)+2 a b (c^6) (α^2)+4 (a^6) (d2^2) (α^2)-8 (a^4) (b^2) (d2^2) (α^2)+4 (a^2) (b^4) (d2^2) (α^2)-8 (a^4) (c^2) (d2^2) (α^2)-8 (a^2) (b^2) (c^2) (d2^2) (α^2)+4 (a^2) (c^4) (d2^2) (α^2)+2 (a^7) b α β+4 (a^6) (b^2) α β-4 (a^4) (b^4) α β+4 (a^2) (b^6) α β+2 a (b^7) α β+2 (a^7) c α β+6 (a^6) b c α β+6 (a^5) (b^2) c α β-2 (a^4) (b^3) c α β-2 (a^3) (b^4) c α β+6 (a^2) (b^5) c α β+6 a (b^6) c α β+2 (b^7) c α β+2 (a^6) (c^2) α β+4 (a^5) b (c^2) α β-2 (a^4) (b^2) (c^2) α β-6 (a^3) (b^3) (c^2) α β-2 (a^2) (b^4) (c^2) α β+4 a (b^5) (c^2) α β+2 (b^6) (c^2) α β-4 (a^5) (c^3) α β-8 (a^4) b (c^3) α β-8 (a^3) (b^2) (c^3) α β-8 (a^2) (b^3) (c^3) α β-8 a (b^4) (c^3) α β-4 (b^5) (c^3) α β-4 (a^4) (c^4) α β-8 (a^3) b (c^4) α β-8 a (b^3) (c^4) α β-4 (b^4) (c^4) α β+2 (a^3) (c^5) α β+6 (a^2) b (c^5) α β+6 a (b^2) (c^5) α β+2 (b^3) (c^5) α β+2 (a^2) (c^6) α β+6 a b (c^6) α β+2 (b^2) (c^6) α β+8 (a^5) b (d2^2) α β-16 (a^3) (b^3) (d2^2) α β+8 a (b^5) (d2^2) α β-16 (a^3) b (c^2) (d2^2) α β-16 a (b^3) (c^2) (d2^2) α β+8 a b (c^4) (d2^2) α β+(a^6) (b^2) (β^2)+2 (a^5) (b^3) (β^2)-2 (a^3) (b^5) (β^2)+2 a (b^7) (β^2)+(b^8) (β^2)+2 (a^6) b c (β^2)+4 (a^5) (b^2) c (β^2)-4 (a^3) (b^4) c (β^2)+2 (a^2) (b^5) c (β^2)+6 a (b^6) c (β^2)+2 (b^7) c (β^2)+2 (a^5) b (c^2) (β^2)-4 (a^3) (b^3) (c^2) (β^2)+(a^2) (b^4) (c^2) (β^2)+2 a (b^5) (c^2) (β^2)-4 (a^4) b (c^3) (β^2)-6 (a^3) (b^2) (c^3) (β^2)-4 (a^2) (b^3) (c^3) (β^2)-4 a (b^4) (c^3) (β^2)-2 (b^5) (c^3) (β^2)-4 (a^3) b (c^4) (β^2)+2 (a^2) b (c^5) (β^2)+4 a (b^2) (c^5) (β^2)+2 (b^3) (c^5) (β^2)+2 a b (c^6) (β^2)+(b^2) (c^6) (β^2)+4 (a^4) (b^2) (d2^2) (β^2)-8 (a^2) (b^4) (d2^2) (β^2)+4 (b^6) (d2^2) (β^2)-8 (a^2) (b^2) (c^2) (d2^2) (β^2)-8 (b^4) (c^2) (d2^2) (β^2)+4 (b^2) (c^4) (d2^2) (β^2)+2 (a^7) b α γ+2 (a^6) (b^2) α γ-4 (a^5) (b^3) α γ-4 (a^4) (b^4) α γ+2 (a^3) (b^5) α γ+2 (a^2) (b^6) α γ+2 (a^7) c α γ+6 (a^6) b c α γ+4 (a^5) (b^2) c α γ-8 (a^4) (b^3) c α γ-8 (a^3) (b^4) c α γ+6 (a^2) (b^5) c α γ+6 a (b^6) c α γ+4 (a^6) (c^2) α γ+6 (a^5) b (c^2) α γ-2 (a^4) (b^2) (c^2) α γ-8 (a^3) (b^3) (c^2) α γ+6 a (b^5) (c^2) α γ+2 (b^6) (c^2) α γ-2 (a^4) b (c^3) α γ-6 (a^3) (b^2) (c^3) α γ-8 (a^2) (b^3) (c^3) α γ-8 a (b^4) (c^3) α γ+2 (b^5) (c^3) α γ-4 (a^4) (c^4) α γ-2 (a^3) b (c^4) α γ-2 (a^2) (b^2) (c^4) α γ-8 a (b^3) (c^4) α γ-4 (b^4) (c^4) α γ+6 (a^2) b (c^5) α γ+4 a (b^2) (c^5) α γ-4 (b^3) (c^5) α γ+4 (a^2) (c^6) α γ+6 a b (c^6) α γ+2 (b^2) (c^6) α γ+2 a (c^7) α γ+2 b (c^7) α γ+8 (a^5) c (d2^2) α γ-16 (a^3) (b^2) c (d2^2) α γ+8 a (b^4) c (d2^2) α γ-16 (a^3) (c^3) (d2^2) α γ-16 a (b^2) (c^3) (d2^2) α γ+8 a (c^5) (d2^2) α γ+2 (a^6) (b^2) β γ+2 (a^5) (b^3) β γ-4 (a^4) (b^4) β γ-4 (a^3) (b^5) β γ+2 (a^2) (b^6) β γ+2 a (b^7) β γ+6 (a^6) b c β γ+6 (a^5) (b^2) c β γ-8 (a^4) (b^3) c β γ-8 (a^3) (b^4) c β γ+4 (a^2) (b^5) c β γ+6 a (b^6) c β γ+2 (b^7) c β γ+2 (a^6) (c^2) β γ+6 (a^5) b (c^2) β γ-8 (a^3) (b^3) (c^2) β γ-2 (a^2) (b^4) (c^2) β γ+6 a (b^5) (c^2) β γ+4 (b^6) (c^2) β γ+2 (a^5) (c^3) β γ-8 (a^4) b (c^3) β γ-8 (a^3) (b^2) (c^3) β γ-6 (a^2) (b^3) (c^3) β γ-2 a (b^4) (c^3) β γ-4 (a^4) (c^4) β γ-8 (a^3) b (c^4) β γ-2 (a^2) (b^2) (c^4) β γ-2 a (b^3) (c^4) β γ-4 (b^4) (c^4) β γ-4 (a^3) (c^5) β γ+4 (a^2) b (c^5) β γ+6 a (b^2) (c^5) β γ+2 (a^2) (c^6) β γ+6 a b (c^6) β γ+4 (b^2) (c^6) β γ+2 a (c^7) β γ+2 b (c^7) β γ+8 (a^4) b c (d2^2) β γ-16 (a^2) (b^3) c (d2^2) β γ+8 (b^5) c (d2^2) β γ-16 (a^2) b (c^3) (d2^2) β γ-16 (b^3) (c^3) (d2^2) β γ+8 b (c^5) (d2^2) β γ+2 (a^6) b c (γ^2)+2 (a^5) (b^2) c (γ^2)-4 (a^4) (b^3) c (γ^2)-4 (a^3) (b^4) c (γ^2)+2 (a^2) (b^5) c (γ^2)+2 a (b^6) c (γ^2)+(a^6) (c^2) (γ^2)+4 (a^5) b (c^2) (γ^2)-6 (a^3) (b^3) (c^2) (γ^2)+4 a (b^5) (c^2) (γ^2)+(b^6) (c^2) (γ^2)+2 (a^5) (c^3) (γ^2)-4 (a^3) (b^2) (c^3) (γ^2)-4 (a^2) (b^3) (c^3) (γ^2)+2 (b^5) (c^3) (γ^2)-4 (a^3) b (c^4) (γ^2)+(a^2) (b^2) (c^4) (γ^2)-4 a (b^3) (c^4) (γ^2)-2 (a^3) (c^5) (γ^2)+2 (a^2) b (c^5) (γ^2)+2 a (b^2) (c^5) (γ^2)-2 (b^3) (c^5) (γ^2)+6 a b (c^6) (γ^2)+2 a (c^7) (γ^2)+2 b (c^7) (γ^2)+(c^8) (γ^2)+4 (a^4) (c^2) (d2^2) (γ^2)-8 (a^2) (b^2) (c^2) (d2^2) (γ^2)+4 (b^4) (c^2) (d2^2) (γ^2)-8 (a^2) (c^4) (d2^2) (γ^2)-8 (b^2) (c^4) (d2^2) (γ^2)+4 (c^6) (d2^2) (γ^2)),
(-(a^2) b d1-a (b^2) d1-(a^2) c d1-a b c d1-(b^2) c d1-a (c^2) d1-b (c^2) d1+(a^3) d2-(a^2) b d2-a (b^2) d2+(b^3) d2-(a^2) c d2+2 a b c d2-(b^2) c d2-a (c^2) d2-b (c^2) d2+(c^3) d2) ((a^2) b d1+a (b^2) d1+(a^2) c d1+a b c d1+(b^2) c d1+a (c^2) d1+b (c^2) d1+(a^3) d2-(a^2) b d2-a (b^2) d2+(b^3) d2-(a^2) c d2+2 a b c d2-(b^2) c d2-a (c^2) d2-b (c^2) d2+(c^3) d2),
(a^4)-2 (a^2) (b^2)+(b^4)-2 (a^2) (c^2)-2 (b^2) (c^2)+(c^4)+4 (a^2) (α^2)+8 a b α β+4 (b^2) (β^2)+8 a c α γ+8 b c β γ+4 (c^2) (γ^2)};

(gb=Factor/@GroebnerBasis[Numerator/@eqns,{α},{d1,d2,β,γ},MonomialOrder->EliminationOrder])//Timing

(Subscript[X, 1924])

CenterName[X[1924]]

f(a,b,c) = a2/b2

CenterFunction[X[1924]]//InputForm

Symmetrize[CenterFunction[X[1924]]]//InputForm

Symmetrize[First[Subtract@@(CyclicTrilinears/@BicentricPair[a^2/b^2])]]//InputForm

Symmetrize[CenterFunction[Brisse[1924]]]//InputForm

(Subscript[X, 2041])

CenterName[X[2041]]

FullSimplify[(b^2 - c^2)^2 - a^2(b^2 + c^2) + 3^(1/2)a^2(b^2 + c^2 - a^2)]

CenterFunction[X[2041]]//InputForm

CenterFunction[Brisse[2041]]//InputForm

Factor[Together//@ToSidelengths[Together//@TrigExpand[First[Intersections@@Line@@@(Map[Trilinears[X[#]]&,{{2,3},{15,485}},{2}])]]]/.Sqrt[x_]:>FullSimplify[Sqrt[x],TriangleQ[]]]

Symmetrize[%]

FullSimplify[%]

InputForm[%]

CenterFunction[Brisse[2041]]

Symmetrize[((CenterFunction[Brisse[2041]])/((-(Sqrt[3]) (a^4)+((((b^2)-(c^2)))^2)+(-1+(Sqrt[3])) (a^2) ((b^2)+(c^2)))/a))]

Symmetrize[((CenterFunction[X[2041]])/((-(Sqrt[3]) (a^4)+((((b^2)-(c^2)))^2)+(-1+(Sqrt[3])) (a^2) ((b^2)+(c^2)))/a))]

(Subscript[X, 2042])

CenterName[X[2042]]

CenterFunction[X[2042]]

CenterFunction[Brisse[2042]]

(Subscript[X, 2043])

CenterName[X[2043]]

CenterFunction[X[2043]]

CenterFunction[Brisse[2043]]

(Subscript[X, 2044])

CenterName[X[2044]]

CenterFunction[X[2044]]

CenterFunction[Brisse[2044]]

(Subscript[X, 2045])

CenterName[X[2045]]

CenterFunction[X[2045]]

CenterFunction[Brisse[2045]]

(Subscript[X, 2046])

CenterName[X[2046]]

CenterFunction[X[2046]]

CenterFunction[Brisse[2046]]

(Subscript[X, 2070])

CenterFunction[X[2070]]

Simplify[Together//@ToSidelengths[CenterFunction[X[2070]]],TriangleQ[]]

Symmetrize[CenterFunction[Brisse[2070]]]

(Subscript[X, 2086])

CenterName[X[2086]]

CenterFunction[X[2086]]//InputForm

Symmetrize[CenterFunction[X[2086]]]//InputForm

Symmetrize[CenterFunction[Brisse[2086]]]//InputForm

Symmetrize[PKTransform[X[99]][[1]]]//InputForm

{2100,"reflection of X(1) in X(1113)",""}

eqn=Symmetrize[Together//@DistanceTrilinears[Reflection[X[1],X[1113]][[1]]]]

{1,√((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))}.Factor/@(Coefficient[eqn,√((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)),#]&/@{0,1})

InputForm[%]

{2101,"reflection of X(1) in X(1114)",""}

eqn=Symmetrize[Together//@DistanceTrilinears[Reflection[X[1],X[1114]][[1]]]]

{1,√((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))}.Factor/@(Coefficient[eqn,√((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)),#]&/@{0,1})

InputForm[%]

{2102,"reflection of X(1113) in X(1)",""}

eqn=Symmetrize[Together//@DistanceTrilinears[Reflection[X[1113],X[1]][[1]]]]

{1,√((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))}.Factor/@(Coefficient[eqn,√((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)),#]&/@{0,1})

InputForm[%]

{2103,"reflection of X(1114) in X(1)",""}

eqn=Symmetrize[Together//@DistanceTrilinears[Reflection[X[1114],X[1]][[1]]]]

{1,√((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))}.Factor/@(Coefficient[eqn,√((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)),#]&/@{0,1})

InputForm[%]

{2104,"reflection of X(1113) in X(6)",""}

eqn=Symmetrize[Together//@DistanceTrilinears[Reflection[X[1113],X[6]][[1]]]]

{1,√((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))}.Factor/@(Coefficient[eqn,√((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)),#]&/@{0,1})

InputForm[%]

{2105,"reflection of X(1114) in X(6)",""}

eqn=Symmetrize[Together//@DistanceTrilinears[Reflection[X[1114],X[6]][[1]]]]

{1,√((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))}.Factor/@(Coefficient[eqn,√((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)),#]&/@{0,1})

InputForm[%]

{2446,1st intersection(line X(1)X(3), incircle),Cos[B]+Cos[C]-1-D; is the closer to X(3).

eqn1=LineEquation[X/@{1,3}]

eqn1=Symmetrize[Numerator[Together//@DistanceTrilinears[LineEquation[X/@{1,3}]]]]

eqn2=CircleEquation[Incircle]

eqn2=Symmetrize[Together[CircleEquation[Incircle]]]

Exact Trilinears

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

x=(-(a^14) b c+(a^13) (b^2) c+5 (a^12) (b^3) c-5 (a^11) (b^4) c-10 (a^10) (b^5) c+10 (a^9) (b^6) c+10 (a^8) (b^7) c-10 (a^7) (b^8) c-5 (a^6) (b^9) c+5 (a^5) (b^10) c+(a^4) (b^11) c-(a^3) (b^12) c+3 (a^13) b (c^2)-9 (a^12) (b^2) (c^2)-5 (a^11) (b^3) (c^2)+34 (a^10) (b^4) (c^2)-10 (a^9) (b^5) (c^2)-46 (a^8) (b^6) (c^2)+30 (a^7) (b^7) (c^2)+24 (a^6) (b^8) (c^2)-25 (a^5) (b^9) (c^2)-(a^4) (b^10) (c^2)+7 (a^3) (b^11) (c^2)-2 (a^2) (b^12) (c^2)+2 (a^12) b (c^3)+12 (a^11) (b^2) (c^3)-33 (a^10) (b^3) (c^3)-17 (a^9) (b^4) (c^3)+80 (a^8) (b^5) (c^3)-20 (a^7) (b^6) (c^3)-62 (a^6) (b^7) (c^3)+42 (a^5) (b^8) (c^3)+6 (a^4) (b^9) (c^3)-16 (a^3) (b^10) (c^3)+7 (a^2) (b^11) (c^3)-a (b^12) (c^3)-14 (a^11) b (c^4)+27 (a^10) (b^2) (c^4)+33 (a^9) (b^3) (c^4)-76 (a^8) (b^4) (c^4)-16 (a^7) (b^5) (c^4)+70 (a^6) (b^6) (c^4)-10 (a^5) (b^7) (c^4)-20 (a^4) (b^8) (c^4)+6 (a^3) (b^9) (c^4)-(a^2) (b^10) (c^4)+a (b^11) (c^4)+5 (a^10) b (c^5)-57 (a^9) (b^2) (c^5)+74 (a^8) (b^3) (c^5)+22 (a^7) (b^4) (c^5)-44 (a^6) (b^5) (c^5)-12 (a^5) (b^6) (c^5)-10 (a^4) (b^7) (c^5)+42 (a^3) (b^8) (c^5)-25 (a^2) (b^9) (c^5)+5 a (b^10) (c^5)+25 (a^9) b (c^6)-18 (a^8) (b^2) (c^6)-74 (a^7) (b^3) (c^6)+84 (a^6) (b^4) (c^6)-44 (a^5) (b^5) (c^6)+70 (a^4) (b^6) (c^6)-62 (a^3) (b^7) (c^6)+24 (a^2) (b^8) (c^6)-5 a (b^9) (c^6)-20 (a^8) b (c^7)+88 (a^7) (b^2) (c^7)-74 (a^6) (b^3) (c^7)+22 (a^5) (b^4) (c^7)-16 (a^4) (b^5) (c^7)-20 (a^3) (b^6) (c^7)+30 (a^2) (b^7) (c^7)-10 a (b^8) (c^7)-20 (a^7) b (c^8)-18 (a^6) (b^2) (c^8)+74 (a^5) (b^3) (c^8)-76 (a^4) (b^4) (c^8)+80 (a^3) (b^5) (c^8)-46 (a^2) (b^6) (c^8)+10 a (b^7) (c^8)+25 (a^6) b (c^9)-57 (a^5) (b^2) (c^9)+33 (a^4) (b^3) (c^9)-17 (a^3) (b^4) (c^9)-10 (a^2) (b^5) (c^9)+10 a (b^6) (c^9)+5 (a^5) b (c^10)+27 (a^4) (b^2) (c^10)-33 (a^3) (b^3) (c^10)+34 (a^2) (b^4) (c^10)-10 a (b^5) (c^10)-14 (a^4) b (c^11)+12 (a^3) (b^2) (c^11)-5 (a^2) (b^3) (c^11)-5 a (b^4) (c^11)+2 (a^3) b (c^12)-9 (a^2) (b^2) (c^12)+5 a (b^3) (c^12)+3 (a^2) b (c^13)+a (b^2) (c^13)-a b (c^14));

Simplify[soln[[1,1,2]],TriangleQ[]]

(2 a b c (Sqrt[Factor[-(a^4)-((((b^2)-(c^2)))^2)+2 (a^2) ((b^2)+(c^2))]]) Factor[((a^6)-(((b-c))^3) c (((b+c))^2)-(a^5) (b+2 c)-(a^4) (2 (b^2)-6 b c+(c^2))+(a^3) (2 (b^3)-2 (b^2) c-5 b (c^2)+4 (c^3))+(a^2) ((b^4)-5 (b^3) c+10 (b^2) (c^2)-5 b (c^3)-(c^4))-a ((b^5)-4 (b^4) c+5 (b^3) (c^2)+2 (b^2) (c^3)-6 b (c^4)+2 (c^5)))]+Factor[(-2 a b c+(a^2) (b+c)-(((b-c))^2) (b+c)) ]√Factor[(-a b c ((a^7)-(a^6) (b+c)+(((b-c))^4) (((b+c))^3)-a ((((b^2)-(c^2)))^2) ((b^2)-3 b c+(c^2))-3 (a^5) ((b^2)-b c+(c^2))+(a^4) (3 (b^3)+(b^2) c+b (c^2)+3 (c^3))-(a^2) (((b-c))^2) (3 (b^3)+5 (b^2) c+5 b (c^2)+3 (c^3))+(a^3) (3 (b^4)-6 (b^3) c+2 (b^2) (c^2)-6 b (c^3)+3 (c^4))))] ((a^3)-a (((b-c))^2)-(a^2) c-(b^2) c+(c^3)))/Factor[(4 a b c ((a^7)-(a^6) c+3 (a^2) (c^3) ((b^2)-(c^2))+c (((-(b^2)+(c^2)))^3)-3 (a^5) ((b^2)-b c+(c^2))+(a^4) (2 (b^2) c+3 (c^3))-a (((b+c))^2) ((b^4)-4 (b^3) c+7 (b^2) (c^2)-5 b (c^3)+(c^4))+(a^3) (3 (b^4)-5 (b^3) c+3 (b^2) (c^2)-6 b (c^3)+3 (c^4))))]

(2 a b c (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) ((a^3)-a (b^2)-(a^2) c+2 a b c-(b^2) c-a (c^2)+(c^3)) ((a^3)-(a^2) b-a (b^2)+(b^3)-(a^2) c+3 a b c-(b^2) c-a (c^2)-b (c^2)+(c^3))+Factor[((a^2) b-(b^3)+(a^2) c-2 a b c+(b^2) c+b (c^2)-(c^3)) ((a^3)-a (((b-c))^2)-(a^2) c-(b^2) c+(c^3))] (Sqrt[a b c])(Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])√ ((a^3)-(a^2) b-a (b^2)+(b^3)-(a^2) c+3 a b c-(b^2) c-a (c^2)-b (c^2)+(c^3)))/(4 a b c (a+b+c) ((a^3)-a (b^2)-(a^2) c+2 a b c-(b^2) c-a (c^2)+(c^3)) ((a^3)-(a^2) b-a (b^2)+(b^3)-(a^2) c+3 a b c-(b^2) c-a (c^2)-b (c^2)+(c^3)))

eqn=Symmetrize[%29]

{1,(Sqrt[(a^3)-(a^2) b-a (b^2)+(b^3)-(a^2) c+3 a b c-(b^2) c-a (c^2)-b (c^2)+(c^3)]) (Sqrt[a b c])}.(Factor/@(Coefficient[eqn,√((a^3)-(a^2) b-a (b^2)+(b^3)-(a^2) c+3 a b c-(b^2) c-a (c^2)-b (c^2)+(c^3)) (Sqrt[a b c]),#]&/@{0,1}))

((a^2) b-(b^3)+(a^2) c-2 a b c+(b^2) c+b (c^2)-(c^3)) +2(Sqrt[a b c]) (Sqrt[(a^3)-(a^2) b-a (b^2)+(b^3)-(a^2) c+3 a b c-(b^2) c-a (c^2)-b (c^2)+(c^3)])

Moses

InputForm[Cos[B]+Cos[C]-1-(Sqrt[3-2Cos[A]-2Cos[B]-2Cos[C]])]

Symmetrize[Together//@DistanceTrilinears[Cos[B]+Cos[C]-1+(Sqrt[3-2Cos[A]-2Cos[B]-2Cos[C]])]]

Simplify[%,TriangleQ[]]

Factor[-(b^3)-2 a b c+(b^2) c+b (c^2)-(c^3)+(a^2) (b+c)]+2 √Factor[a b c ((a^3)-(a^2) (b+c)+(((b-c))^2) (b+c)-a ((b^2)-3 b c+(c^2)))]

{2447,"2nd intersection(line X(1)X(3), incircle)","Cos[B]+Cos[C]-1+D"}

InputForm[Cos[B]+Cos[C]-1+(Sqrt[3-2Cos[A]-2Cos[B]-2Cos[C]])]

(Subscript[X, 2476])

CenterName[X[2476]]

Together//@ToSidelengths[InversePoint[Trilinears[X[21]],Circle[OrthocentroidalCircle]]/.Cos[x_]:>TrigExpand[Cos[x]]]

Factor[%/.Sqrt[x_]Sqrt[y_]:>Simplify[Sqrt[x y],TriangleQ[]]]

Symmetrize[-First[%24]]

Symmetrize[CenterFunction[X[2476]]]

CenterFunction[Brisse[2476]]

(Subscript[X, 2478])

CenterName[X[2478]]

Together//@ToSidelengths[InversePoint[Trilinears[X[377]],Circle[OrthocentroidalCircle]]/.Cos[x_]:>TrigExpand[Cos[x]]]

Symmetrize[First[%/.Sqrt[x_]Sqrt[y_]:>Simplify[Sqrt[x y],TriangleQ[]]]]

Symmetrize[CenterFunction[X[2478]]]

CenterFunction[Brisse[2478]]

(Subscript[X, 2481])

CenterName[X[2481]]

feuerbach=((Cos[A]-Cos[B])/γ)+((Cos[B]-Cos[C])/α)+((-Cos[A]+Cos[C])/β);
steiner=(1/(a α))+(1/(b β))+(1/(c γ));

(Last/@Flatten[DistanceTrilinears[Solve[{feuerbach==0,steiner==0,γ==1},{α,β}]]]//Simplify//Factor)(1/((c^2) ((a^2)+(b^2)-a c-b c)))//Simplify//Factor

-(1/(a^2*(a*b - b^2 + a*c - c^2)))

CenterFunction[X[2481]]

-CenterFunction[Brisse[2481]]//Factor

(Subscript[X, 2482])

CenterName[X[2482]]

CenterFunction[X[2482]]

CenterFunction[Brisse[2482]]

ConicEquation[SteinerInellipse]

solns=ExactTrilinears/@({α,β,γ}/.(Together//@Solve[LineEquation[Line[X[115],Centroid]]==0&&ConicEquation[SteinerInellipse]==0&&γ==1,{α,β,γ}]))

Symmetrize/@Trilinears[X[115]]

Symmetrize/@solns[[1]]

Symmetrize/@solns[[2]]

Factor[ConicEquation[SteinerInellipse]/.Thread[{α,β,γ}->CyclicTrilinears[(((2 (a^2)-(b^2)-(c^2)))^2)]]]

Factor[ConicEquation[SteinerInellipse]/.Thread[{α,β,γ}->CyclicTrilinears[(((2 (a^2)-(b^2)-(c^2)))^2)/a]]]

(Subscript[X, 2659])

CenterName[X[2659]]

1/(sec C + sec A)

Crossdifference[tri1_?VectorQ,tri2_?VectorQ]:=Cross[tri1,tri2]
Crossdifference[ctr1_,ctr2_]:=Crossdifference@@(Trilinears/@{ctr1,ctr2})/;And@@(MemberQ[Centers,#]&/@{ctr1,ctr2})

et $P=p:q:r$ and $U=u:v:w$ be distinct points,neither lying on a sideline of $\Delta ABC$.Then the
crossdifference of $P$ and $U$ is the point $X$ defined by trilinears
$$qw-rv:ru-pw:pv-qu\mathpunct{.}$$
Null

Simplify[CenterFunction[X[2659]]]

Simplify[Crossdifference@@(CyclicTrilinears/@BicentricPair[1/(Sec[C]+Sec[A])])][[1]]

Symmetrize[Simplify[Together//@DistanceTrilinears[%],TriangleQ[]]]

Symmetrize[CenterFunction[Brisse[2659]]]

(Subscript[X, 2479]) (1st INTERSECTION(EULER LINE, STEINER CIRCUMELLIPSE, the one closer toX(3).)

Solve[{ConicEquation[SteinerCircumellipse]==0,LineEquation[EulerLine]==0,4Area[]^2==(((a α+b β+c γ))^2)},{α},{β,γ}]

FullSimplify[Together//@DistanceTrilinears[Solve[{ConicEquation[SteinerCircumellipse]==0,LineEquation[EulerLine]==0,γ==1},{α},{β}][[1,1,2]]],TriangleQ[]]

Factor[(a^8)+(b^8)+(a^4) (b^2) (c^2)-(b^6) (c^2)-(b^2) (c^6)+(c^8)-(a^6) ((b^2)+(c^2))-(a^2) ((((b^2)-(c^2)))^2) ((b^2)+(c^2))]

K - 3a2bc cos B cos C + 4(area ABC)2,      K as at X(2454)

(Subscript[X, 2480]) (2nd INTERSECTION(EULER LINE, STEINER CIRCUMELLIPSE, the one farther from X(3).)

(Subscript[X, 2574])

Symmetrize[FullSimplify[Together//@DistanceTrilinears[IsogonalConjugate[X[1113]][[1]]],TriangleQ[]]]

1/({1,(Sqrt[(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)])}.Factor/@(Coefficient[(-2 (a^4) b c+(a^2) (b^3) c+(b^5) c+(a^2) b (c^3)-2 (b^3) (c^3)+b (c^5)+(a^3) (Sqrt[(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)])-a (b^2) (Sqrt[(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)])-a (c^2) (Sqrt[(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)])),(Sqrt[(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)]),#]&/@{0,1}))

InputForm[%]

(Subscript[X, 2575])

Symmetrize[FullSimplify[Together//@DistanceTrilinears[IsogonalConjugate[X[1114]][[1]]],TriangleQ[]]]

-1/({1,(Sqrt[(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)])}.Factor/@(Coefficient[2 (a^4) b c-(a^2) (b^3) c-(b^5) c-(a^2) b (c^3)+2 (b^3) (c^3)-b (c^5)+(a^3) (Sqrt[(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)])-a (b^2) (Sqrt[(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)])-a (c^2) (Sqrt[(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)]),(Sqrt[(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)]),#]&/@{0,1}))

InputForm[%]

(Subscript[X, 2645])

X(2645) = MIDPOINT OF PU(79)

From definition in BicentricPairs

PU 79 =1/(cot C-cot A)

(Together//@ToSidelengths[Midpoint[CyclicTrilinears/@BicentricPair[(1/(Cot[C]-Cot[A]))]]])/.Sqrt[x_]:>Refine[Sqrt[x],TriangleQ[]]

Factor[%/.Sqrt[x_]:>Refine[Sqrt[x],TriangleQ[]]]

Symmetrize/@%

First[%]

Based on ETC's formula

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = k(a,b,c)/(cot C - cot A) + h(a,b,c)/(cot B - cot A),
                        where h(a,b,c) = a/(cot C - cot A) + b/(cot A - cot B) + c/(cot B - cot C),
                           and k(a,b,c) = a/(cot B - cot A) + b/(cot C - cot B) + c/(cot A - cot C)

h:=(a/(Cot[C]-Cot[A]))+(b/(Cot[A]-Cot[B]))+(c/(Cot[B]-Cot[C]))

k:=(a/(Cot[B]-Cot[A]))+(b/(Cot[C]-Cot[B]))+(c/(Cot[A]-Cot[C]))

Factor[Together//@((k/(Cot[C]-Cot[A]))+(h/(Cot[B]-Cot[A])))]

(Together//@DistanceTrilinears[Expand[(Cot[B]-Cot[C])(b ((Cot[A])^3)-c ((Cot[A])^3)+2 a ((Cot[A])^2) Cot[B]-3 b ((Cot[A])^2) Cot[B]-2 a Cot[A] ((Cot[B])^2)+b Cot[A] ((Cot[B])^2)+2 c Cot[A] ((Cot[B])^2)-c ((Cot[B])^3)-2 a ((Cot[A])^2) Cot[C]+3 c ((Cot[A])^2) Cot[C]+4 b Cot[A] Cot[B] Cot[C]-4 c Cot[A] Cot[B] Cot[C]+2 a ((Cot[B])^2) Cot[C]-b ((Cot[B])^2) Cot[C]+c ((Cot[B])^2) Cot[C]+2 a Cot[A] ((Cot[C])^2)-2 b Cot[A] ((Cot[C])^2)-c Cot[A] ((Cot[C])^2)-2 a Cot[B] ((Cot[C])^2)-b Cot[B] ((Cot[C])^2)+c Cot[B] ((Cot[C])^2)+b ((Cot[C])^3))]])/. Sqrt[x_]:>Refine[Sqrt[x],TriangleQ[]]

Factor[Symmetrize[%23]]

InputForm[%]

Symmetrize[CenterFunction[Brisse[2645]]]//InputForm

Together//@Midpoint[CyclicTrilinears/@BicentricPair[1/(Cot[C] - Cot[A])]][[1]]

Factor[Together//@((k/(Cot[C]-Cot[A]))+(h/(Cot[B]-Cot[A])))]

(Subscript[X, 2649])

X(2649)=MIDPOINT OF PU(80)
Trilinears f(A,B,C):f(B,C,A):f(C,A,B),where f(A,B,C)=k(a,b,c)(cos C+cos A)+h(a,b,c)(cos B+cos A),where h(a,b,c)=a(cos C+cos A)+b(cos A+cos B)+c(cos B+cos C),and k(a,b,c)=a(cos B+cos A)+b(cos C+cos B)+c(cos A+cos C)

h:=a(Cos[C]+Cos[A])+b(Cos[A]+Cos[B])+c(Cos[B]+Cos[C])

k:=a(Cos[B]+Cos[A])+b(Cos[C]+Cos[B])+c(Cos[A]+Cos[C])

eqn1=Factor[Together//@(k(Cos[C]+Cos[A])+h(Cos[B]+Cos[A]))]

eqn2=Together//@Midpoint[CyclicTrilinears/@BicentricPair[(Cos[C] + Cos[A])]][[1]]

eqn1==eqn2

Symmetrize[Factor[(Together//@DistanceTrilinears[eqn1])/. Sqrt[x_]:>Refine[Sqrt[x],TriangleQ[]]]]

InputForm[%]

Symmetrize[CenterFunction[Brisse[2649]]]//InputForm

(Subscript[X, 2666])

X(2666) = MIDPOINT OF PU(84)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = k(a,b,c)/(1/c + 1/a) + h(a,b,c)/(1/b + 1/a),
                        where h(a,b,c) = 3 + b/a + c/b + a/c,    k(a,b,c) = 3 + a/b + b/c + c/a

h:=3+b/a+c/b+a/c
k:=3+a/b+b/c+c/a

Symmetrize[Together//@((k/((1/c)+(1/a)))+(h/((1/b)+(1/a))))]//InputForm

Symmetrize[Together//@Midpoint[CyclicTrilinears/@BicentricPair[1/c+1/a]][[1]]]//InputForm

Symmetrize[Together//@ExactTrilinears[Midpoint[CyclicTrilinears/@BicentricPair[1/c+1/a]]][[1]]]

Symmetrize[CenterFunction[Brisse[2666]]]//InputForm

Plot

First 110

t=Triangle[{{0,0},{1,0},{.2,.41}}];
npts=110;
Show[Graphics[{
t,
PointSize[.02],
Table[{
Hue[(n-1)/npts],
Text[TraditionalForm[Subscript[X,n]],Coordinates@X[n][t]+{.03,.05}],
X[n][t]
},{n,npts}]
}],AspectRatio->Automatic]

All

npts=Length[ctr=First/@Select[Centers,MatchQ[#,_X]&&!StringQ[CenterFunction[#]]&]]

Off[Power::"infy"];
Off[∞::"indet"];
Off[Graphics::"gptn"];

t=Triangle[{{0,0},{1,0},{.2,.41}}];
Show[Graphics[{
PointSize[.015],
Table[{
Hue[(n-1)/npts],
Text[TraditionalForm[Subscript[X,ctr[[n]]]],Coordinates@X[ctr[[n]]][t]+{.03,.05}],
X[ctr[[n]]][t]
},{n,npts}],
t,
PointLabels[Transpose[{VertexPoints[t],{"A","B","C"}}],{LabelOffset->.03}]
}],AspectRatio->Automatic,ImageSize->500]