Taylor Circle

Author

Eric W. Weisstein
November 29, 2004

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

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

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

Figure

t=Triangle[{{0,0},{1,.1},{.3,.75}}];
Show[Graphics[{
t,
{Purple,Line/@Transpose[{Vertices[OrthicTriangle[t]],#}]&/@Transpose[TaylorHexagon[t]],
PerpSquare[#,.035,4]&/@Transpose[{Flatten[Line/@Transpose[{Vertices[OrthicTriangle[t]],#}]&/@Transpose[TaylorHexagon[t]]],Line[Vertices[t][[#]]]&/@{{1,2},{1,2},{1,3},{1,3},{2,3},{2,3}}}]
},
{Red,Line/@TaylorHexagon[t]},
{DarkGreen,TaylorCircle[t]},
{Blue,OrthicTriangle[t],Dashing[{.02}],Altitudes[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[OrthicTriangle[t]],{"(Subscript[H, A])","(Subscript[H, B])","(Subscript[H, C])"}}],
Append[#,Red]&/@Flatten[Transpose/@Transpose[{Map[Point,TaylorHexagon[t],{2}],{{"(Subscript[P, A])","(Subscript[Q, A])"},Reverse@{"(Subscript[P, B])","(Subscript[Q, B])"},{"(Subscript[P, C])","(Subscript[Q, C])"}}}],1],
{
{TaylorCenter[t],"(Subscript[X, 389])",DarkGreen},
{Orthocenter[t],"H",Blue}
}
],LabelOffset->.03,PointSize[.025]
]
},AspectRatio->Automatic,PlotRange->All]
]

{Distance@@@(Coordinates/@Line/@TaylorHexagon[t]),2Circumradius[t]Times@@(Sin/@Angles[t]),
Plus@@Distance@@@Partition[Vertices[OrthicTriangle[t]],2,1,1]/2}

Theorems

tri={{0,0},{1,.1},{.3,.9}};
alttri=Table[AltitudeFeet[tri][[i,1]],{i,3}];
g=PerpendicularFeet[alttri,tri];

Show[Graphics[{
			Triangle[tri],Altitudes[tri],
		{Dashing[{.02,.02}],Line/@Perpendiculars[alttri,tri]},
(*			Table[Text[i,g[[i]]+{0,.05}],{i,Length[g]}], *)
			Table[Circumcircle[{g[[2i-1]],g[[2i]],alttri[[i]]}],{i,3}],
			{PointSize[.03],AltitudeFeet[tri]},
			{PointSize[.03],GrayLevel[.5],Point/@PerpendicularFeet[alttri,tri]}
		}],AspectRatio->Automatic]

Show[Graphics[{
			Triangle[tri],Altitudes[tri],
		{Dashing[{.02,.02}],Line/@Perpendiculars[alttri,tri]},
(*			Table[Text[i,g[[i]]+{0,.05}],{i,Length[g]}], *)
			{Circumcircle[{alttri[[2]],alttri[[3]],g[[4]]}]},
			{Circumcircle[{alttri[[1]],alttri[[3]],g[[1]]}]},
			{Circumcircle[{alttri[[1]],alttri[[2]],g[[3]]}]},
(*			{Thickness[.02],Circumcircle[{alttri[[3]],alttri[[2]],g[[3]]}]}, *)
			{PointSize[.03],AltitudeFeet[tri]},
			{PointSize[.03],GrayLevel[.5],Point/@PerpendicularFeet[alttri,tri]}
		}],AspectRatio->Automatic]

Show[Graphics[{
			Triangle[tri],Altitudes[tri],
		{Dashing[{.02,.02}],Line/@Perpendiculars[alttri,tri]},
(*			Table[Text[i,g[[i]]+{0,.05}],{i,Length[g]}], *)
			{Circumcircle[{tri[[1]],g[[4]],g[[5]]}]},
				{Circumcircle[{tri[[2]],g[[1]],g[[6]]}]},
				{Circumcircle[{tri[[3]],g[[2]],g[[3]]}]},
			{PointSize[.03],AltitudeFeet[tri]},
			{PointSize[.03],GrayLevel[.5],Point/@PerpendicularFeet[alttri,tri]}
		}],AspectRatio->Automatic]

o=Orthocenter[tri][[1]];
Show[Graphics[{
			Triangle[tri],Altitudes[tri],
		{Dashing[{.02,.02}],Line/@Perpendiculars[alttri,tri]},
(*			Table[Text[i,g[[i]]+{0,.05}],{i,Length[g]}], *)
			{Circumcircle[{o,g[[1]],g[[4]]}]},
			{Circumcircle[{o,g[[2]],g[[5]]}]},
			{Circumcircle[{o,g[[3]],g[[6]]}]},
			{PointSize[.03],AltitudeFeet[tri]},
			{PointSize[.03],GrayLevel[.5],Point/@PerpendicularFeet[alttri,tri]}
		}],AspectRatio->Automatic]

Equation

CircleEquation[TaylorCircle]

Circle Function

CircleFunction[TaylorCircle]

Kimberling

IncidentCenters[KimberlingCenters,CyclicTrilinears[CircleFunction[TaylorCircle]]]

CenterName[X[394]]

Derivation

o=TrilinearVertexMatrix[OrthicTriangle]

Intersections[PerpendicularLine[o[[1]],LineEquation[#]],#]&/@{{{1,0,0},{0,1,0}},{{1,0,0},{0,0,1}}}

Intersections[PerpendicularLine[o[[2]],LineEquation[#]],#]&/@{{{1,0,0},{0,1,0}},{{0,1,0},{0,0,1}}}

Intersections[PerpendicularLine[o[[3]],LineEquation[#]],#]&/@{{{1,0,0},{0,0,1}},{{0,1,0},{0,0,1}}}

t=Triangle[{{0,0},{1,1/10},{2/10,3/4}}];
Show[Graphics[{
t,
OrthicTriangle[t],
TaylorCircle[t],
PointSize[.02],
Point[TrilinearToCartesian[t,#/.Thread[{A,B,C}->Angles[t]]]]&/@Join[
{{-Cos[B] Sec[C],-Sec[B]-Cos[A] Sec[C],0},{-Cos[C] Sec[B],0,-Cos[A] Sec[B]-Sec[C]}},
{{-Sec[A]-Cos[B] Sec[C],-Cos[A] Sec[C],0},{0,-Cos[C] Sec[A],-Cos[B] Sec[A]-Sec[C]}},
{{-Sec[A]-Cos[C] Sec[B],0,-Cos[A] Sec[B]},{0,-Cos[C] Sec[A]-Sec[B],-Cos[B] Sec[A]}}
]
}],AspectRatio->Automatic]

t=Triangle[{{0,0},{1,.1},{.2,.75}}];
Show[Graphics[{
t,
OrthicTriangle[t],
TaylorCircle[t],
PointSize[.02],
Point[TrilinearToCartesian[t,#/.Thread[{A,B,C}->Angles[t]]]]&/@
{{-Cos[B] Sec[C],-Sec[B]-Cos[A] Sec[C],0},{-Cos[C] Sec[B],0,-Cos[A] Sec[B]-Sec[C]}}
}],AspectRatio->Automatic]

Factor[Together//@DistanceTrilinears[CircleEquation[{{-Cos[B] Sec[C],-Sec[B]-Cos[A] Sec[C],0},{-Cos[C] Sec[B],0,-Cos[A] Sec[B]-Sec[C]},
{-Sec[A]-Cos[B] Sec[C],-Cos[A] Sec[C],0}}]]]

FullSimplify[CircleFunction[(a^10) (α^2)-4 (a^8) (b^2) (α^2)+6 (a^6) (b^4) (α^2)-4 (a^4) (b^6) (α^2)+(a^2) (b^8) (α^2)-4 (a^8) (c^2) (α^2)+8 (a^6) (b^2) (c^2) (α^2)-4 (a^4) (b^4) (c^2) (α^2)+6 (a^6) (c^4) (α^2)-4 (a^4) (b^2) (c^4) (α^2)-2 (a^2) (b^4) (c^4) (α^2)-4 (a^4) (c^6) (α^2)+(a^2) (c^8) (α^2)+2 (a^9) b α β-8 (a^7) (b^3) α β+12 (a^5) (b^5) α β-8 (a^3) (b^7) α β+2 a (b^9) α β-4 (a^7) b (c^2) α β+4 (a^5) (b^3) (c^2) α β+4 (a^3) (b^5) (c^2) α β-4 a (b^7) (c^2) α β+4 (a^5) b (c^4) α β+8 (a^3) (b^3) (c^4) α β+4 a (b^5) (c^4) α β-4 (a^3) b (c^6) α β-4 a (b^3) (c^6) α β+2 a b (c^8) α β+(a^8) (b^2) (β^2)-4 (a^6) (b^4) (β^2)+6 (a^4) (b^6) (β^2)-4 (a^2) (b^8) (β^2)+(b^10) (β^2)-4 (a^4) (b^4) (c^2) (β^2)+8 (a^2) (b^6) (c^2) (β^2)-4 (b^8) (c^2) (β^2)-2 (a^4) (b^2) (c^4) (β^2)-4 (a^2) (b^4) (c^4) (β^2)+6 (b^6) (c^4) (β^2)-4 (b^4) (c^6) (β^2)+(b^2) (c^8) (β^2)+2 (a^9) c α γ-4 (a^7) (b^2) c α γ+4 (a^5) (b^4) c α γ-4 (a^3) (b^6) c α γ+2 a (b^8) c α γ-8 (a^7) (c^3) α γ+4 (a^5) (b^2) (c^3) α γ+8 (a^3) (b^4) (c^3) α γ-4 a (b^6) (c^3) α γ+12 (a^5) (c^5) α γ+4 (a^3) (b^2) (c^5) α γ+4 a (b^4) (c^5) α γ-8 (a^3) (c^7) α γ-4 a (b^2) (c^7) α γ+2 a (c^9) α γ+2 (a^8) b c β γ-4 (a^6) (b^3) c β γ+4 (a^4) (b^5) c β γ-4 (a^2) (b^7) c β γ+2 (b^9) c β γ-4 (a^6) b (c^3) β γ+8 (a^4) (b^3) (c^3) β γ+4 (a^2) (b^5) (c^3) β γ-8 (b^7) (c^3) β γ+4 (a^4) b (c^5) β γ+4 (a^2) (b^3) (c^5) β γ+12 (b^5) (c^5) β γ-4 (a^2) b (c^7) β γ-8 (b^3) (c^7) β γ+2 b (c^9) β γ+(a^8) (c^2) (γ^2)-2 (a^4) (b^4) (c^2) (γ^2)+(b^8) (c^2) (γ^2)-4 (a^6) (c^4) (γ^2)-4 (a^4) (b^2) (c^4) (γ^2)-4 (a^2) (b^4) (c^4) (γ^2)-4 (b^6) (c^4) (γ^2)+6 (a^4) (c^6) (γ^2)+8 (a^2) (b^2) (c^6) (γ^2)+6 (b^4) (c^6) (γ^2)-4 (a^2) (c^8) (γ^2)-4 (b^2) (c^8) (γ^2)+(c^10) (γ^2)]]

Simplify[LawOfCosines[(((a-b-c) (a+b-c) (a-b+c) (a+b+c) (((-(a^2)+(b^2)+(c^2)))^2))/(16 (a^2) (b^3) (c^3)))]Circumradius[]^2/R^2]

Center

CenterName[X[389]]

Spieker circle

t

SpiekerCircle[OrthicTriangle[t]]

X[389][t]

Kimberling

IncidentCenters[KimberlingCenters,CircleCenter[TaylorCircle]]

CenterFunction[X[389]]

Derivation

FullSimplify[Together//@DistanceTrilinears[CircleCenter[TaylorCircle]]]

Factor[%]

Radius

From Tucker Circle Equation

Factor[Together//@DistanceTrilinears[Solve[λ^2+(1-λ)^2Tan[BrocardAngle[]]^2==((Cos[A])^2) ((Cos[B])^2) ((Cos[C])^2)+((Sin[A])^2) ((Sin[B])^2) ((Sin[C])^2),λ]]]

(1/(8 ((a^2) (b^2)+(a^2) (c^2)+(b^2) (c^2))a^2 b^2 c^2))(Expand[(-2 (a^4)+4 (a^2) (b^2)-2 (b^4)+4 (a^2) (c^2)+4 (b^2) (c^2)-2 (c^4))(a^2 b^2 c^2)]-√((a^16) (b^4)-2 (a^14) (b^6)-(a^12) (b^8)+4 (a^10) (b^10)-(a^8) (b^12)-2 (a^6) (b^14)+(a^4) (b^16)+2 (a^16) (b^2) (c^2)-12 (a^12) (b^6) (c^2)+10 (a^10) (b^8) (c^2)+10 (a^8) (b^10) (c^2)-12 (a^6) (b^12) (c^2)+2 (a^2) (b^16) (c^2)+(a^16) (c^4)-13 (a^12) (b^4) (c^4)-4 (a^10) (b^6) (c^4)+32 (a^8) (b^8) (c^4)-4 (a^6) (b^10) (c^4)-13 (a^4) (b^12) (c^4)+(b^16) (c^4)-2 (a^14) (c^6)-12 (a^12) (b^2) (c^6)-4 (a^10) (b^4) (c^6)+18 (a^8) (b^6) (c^6)+18 (a^6) (b^8) (c^6)-4 (a^4) (b^10) (c^6)-12 (a^2) (b^12) (c^6)-2 (b^14) (c^6)-(a^12) (c^8)+10 (a^10) (b^2) (c^8)+32 (a^8) (b^4) (c^8)+18 (a^6) (b^6) (c^8)+32 (a^4) (b^8) (c^8)+10 (a^2) (b^10) (c^8)-(b^12) (c^8)+4 (a^10) (c^10)+10 (a^8) (b^2) (c^10)-4 (a^6) (b^4) (c^10)-4 (a^4) (b^6) (c^10)+10 (a^2) (b^8) (c^10)+4 (b^10) (c^10)-(a^8) (c^12)-12 (a^6) (b^2) (c^12)-13 (a^4) (b^4) (c^12)-12 (a^2) (b^6) (c^12)-(b^8) (c^12)-2 (a^6) (c^14)-2 (b^6) (c^14)+(a^4) (c^16)+2 (a^2) (b^2) (c^16)+(b^4) (c^16)));

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

Factor[((a^16) (b^4)-2 (a^14) (b^6)-(a^12) (b^8)+4 (a^10) (b^10)-(a^8) (b^12)-2 (a^6) (b^14)+(a^4) (b^16)+2 (a^16) (b^2) (c^2)-12 (a^12) (b^6) (c^2)+10 (a^10) (b^8) (c^2)+10 (a^8) (b^10) (c^2)-12 (a^6) (b^12) (c^2)+2 (a^2) (b^16) (c^2)+(a^16) (c^4)-13 (a^12) (b^4) (c^4)-4 (a^10) (b^6) (c^4)+32 (a^8) (b^8) (c^4)-4 (a^6) (b^10) (c^4)-13 (a^4) (b^12) (c^4)+(b^16) (c^4)-2 (a^14) (c^6)-12 (a^12) (b^2) (c^6)-4 (a^10) (b^4) (c^6)+18 (a^8) (b^6) (c^6)+18 (a^6) (b^8) (c^6)-4 (a^4) (b^10) (c^6)-12 (a^2) (b^12) (c^6)-2 (b^14) (c^6)-(a^12) (c^8)+10 (a^10) (b^2) (c^8)+32 (a^8) (b^4) (c^8)+18 (a^6) (b^6) (c^8)+32 (a^4) (b^8) (c^8)+10 (a^2) (b^10) (c^8)-(b^12) (c^8)+4 (a^10) (c^10)+10 (a^8) (b^2) (c^10)-4 (a^6) (b^4) (c^10)-4 (a^4) (b^6) (c^10)+10 (a^2) (b^8) (c^10)+4 (b^10) (c^10)-(a^8) (c^12)-12 (a^6) (b^2) (c^12)-13 (a^4) (b^4) (c^12)-12 (a^2) (b^6) (c^12)-(b^8) (c^12)-2 (a^6) (c^14)-2 (b^6) (c^14)+(a^4) (c^16)+2 (a^2) (b^2) (c^16)+(b^4) (c^16))]

FullSimplify[((a^6) (b^2)-2 (a^4) (b^4)+(a^2) (b^6)+(a^6) (c^2)-(a^4) (b^2) (c^2)-(a^2) (b^4) (c^2)+(b^6) (c^2)-2 (a^4) (c^4)-(a^2) (b^2) (c^4)-2 (b^4) (c^4)+(a^2) (c^6)+(b^2) (c^6))>0,TriangleQ[]]

LawOfCosines[Factor[(1/(8 ((a^2) (b^2)+(a^2) (c^2)+(b^2) (c^2))a^2 b^2 c^2))(-2 (a^2) (b^2) (a-b-c) (a+b-c) (c^2) (a-b+c) (a+b+c)+((a^2)+(b^2)+(c^2)) ((a^6) (b^2)-2 (a^4) (b^4)+(a^2) (b^6)+(a^6) (c^2)-(a^4) (b^2) (c^2)-(a^2) (b^4) (c^2)+(b^6) (c^2)-2 (a^4) (c^4)-(a^2) (b^2) (c^4)-2 (b^4) (c^4)+(a^2) (c^6)+(b^2) (c^6)))]]

eqn1=Factor[Together//@DistanceTrilinears[λ^2+(1-λ)^2Tan[BrocardAngle[]]^2/.λ->-Cos[A] Cos[B] Cos[C]]]

eqn2=Together//@DistanceTrilinears[Sin[A]^2Sin[B]^2Sin[C]^2+Cos[A]^2Cos[B]^2Cos[C]^2]

eqn1-eqn2

Grinberg's Form

Solve[-Cos[A]Cos[B]Cos[C]==tanw/(tanw-Tan[A]Tan[B]Tan[C]),tanw]

Derivation

Together//@DistanceTrilinears[CircleRadius[TaylorCircle]]

Together//@(Together//@DistanceTrilinears[Distance[{-Cos[B] Sec[C],-Sec[B]-Cos[A] Sec[C],0},TrigExpand[Trilinears[X[389]]]]]/.(Sqrt[-(((a-b-c) (a+b-c) (a-b+c) (a+b+c))/(((x_)^2) ((y_)^2)))]):>sqrt/(x y)/.sqrt^2->(-a+b+c) (a+b-c) (a-b+c) (a+b+c))

FullSimplify[%32 R/Circumradius[],TriangleQ[]]

(R/(4 (a^2) (b^2) (c^2)))√((b^2) (c^2) ((((b^2)-(c^2)))^4)+(a^10) ((b^2)+(c^2))+(a^2) ((((b^2)-(c^2)))^4) ((b^2)+(c^2))+(a^8) (-4 (b^4)-3 (b^2) (c^2)-4 (c^4))-2 (a^4) ((((b^2)-(c^2)))^2) (2 (b^4)+3 (b^2) (c^2)+2 (c^4))+2 (a^6) (3 (b^6)+(b^4) (c^2)+(b^2) (c^4)+3 (c^6)))

Factor[((b^2) (c^2) ((((b^2)-(c^2)))^4)+(a^10) ((b^2)+(c^2))+(a^2) ((((b^2)-(c^2)))^4) ((b^2)+(c^2))+(a^8) (-4 (b^4)-3 (b^2) (c^2)-4 (c^4))-2 (a^4) ((((b^2)-(c^2)))^2) (2 (b^4)+3 (b^2) (c^2)+2 (c^4))+2 (a^6) (3 (b^6)+(b^4) (c^2)+(b^2) (c^4)+3 (c^6))) ]

t=Triangle[{{0,0},{1,1/10},{2/10,3/4}}];

N[TaylorCircle[t],50]

N[(R/(4 (a^2) (b^2) (c^2)))√((b^2) (c^2) ((((b^2)-(c^2)))^4)+(a^10) ((b^2)+(c^2))+(a^2) ((((b^2)-(c^2)))^4) ((b^2)+(c^2))+(a^8) (-4 (b^4)-3 (b^2) (c^2)-4 (c^4))-2 (a^4) ((((b^2)-(c^2)))^2) (2 (b^4)+3 (b^2) (c^2)+2 (c^4))+2 (a^6) (3 (b^6)+(b^4) (c^2)+(b^2) (c^4)+3 (c^6))) /.R->Circumradius[t]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]],50]

Expand[((b^2) (c^2) ((((b^2)-(c^2)))^4)+(a^10) ((b^2)+(c^2))+(a^2) ((((b^2)-(c^2)))^4) ((b^2)+(c^2))+(a^8) (-4 (b^4)-3 (b^2) (c^2)-4 (c^4))-2 (a^4) ((((b^2)-(c^2)))^2) (2 (b^4)+3 (b^2) (c^2)+2 (c^4))+2 (a^6) (3 (b^6)+(b^4) (c^2)+(b^2) (c^4)+3 (c^6))) ]

eqn=(a^10) (b^2)-4 (a^8) (b^4)+6 (a^6) (b^6)-4 (a^4) (b^8)+(a^2) (b^10)+(a^10) (c^2)-3 (a^8) (b^2) (c^2)+2 (a^6) (b^4) (c^2)+2 (a^4) (b^6) (c^2)-3 (a^2) (b^8) (c^2)+(b^10) (c^2)-4 (a^8) (c^4)+2 (a^6) (b^2) (c^4)+4 (a^4) (b^4) (c^4)+2 (a^2) (b^6) (c^4)-4 (b^8) (c^4)+6 (a^6) (c^6)+2 (a^4) (b^2) (c^6)+2 (a^2) (b^4) (c^6)+6 (b^6) (c^6)-4 (a^4) (c^8)-3 (a^2) (b^2) (c^8)-4 (b^4) (c^8)+(a^2) (c^10)+(b^2) (c^10);

16a^4b^4c^4Together//@DistanceTrilinears[Sin[A]^2Sin[B]^2Sin[C]^2+Cos[A]^2Cos[B]^2Cos[C]^2]-eqn

Symmetry Check

<<Algebra`SymmetricPolynomials`

SymmetricReduction[eqn,{a,b,c}]

Groebner

cos=CyclicTrilinears[Numerator[Together[#-DistanceTrilinears[#]&[Cos[A]Cos[B]Cos[C]]]]]

eqn

Factor[GroebnerBasis[{eqn,Sequence@@cos},{a,b,c},MonomialOrder->EliminationOrder]]

eqn

eqn1=-(a^12)+9 (a^8) (b^4)-16 (a^6) (b^6)+9 (a^4) (b^8)-(b^12)+9 (a^8) (c^4)-18 (a^4) (b^4) (c^4)+9 (b^8) (c^4)-16 (a^6) (c^6)-16 (b^6) (c^6)+9 (a^4) (c^8)+9 (b^4) (c^8)-(c^12)+64 (a^4) (b^4) (c^4) ((Cos[A])^2) ((Cos[B])^2) ((Cos[C])^2)

f1=-Factor[DistanceTrilinears[eqn1]/eqn]

id=(-(a^12)+9 (a^8) (b^4)-16 (a^6) (b^6)+9 (a^4) (b^8)-(b^12)+9 (a^8) (c^4)-18 (a^4) (b^4) (c^4)+9 (b^8) (c^4)-16 (a^6) (c^6)-16 (b^6) (c^6)+9 (a^4) (c^8)+9 (b^4) (c^8)-(c^12))

sin=CyclicTrilinears[Numerator[Together[#-DistanceTrilinears[#]&[(Sin[A]Sin[B]Sin[C])^2]]]]

Factor[GroebnerBasis[{id,Sequence@@sin},{a,b,c},MonomialOrder->EliminationOrder]]

eqn2=-3 (a^10) (b^2)+12 (a^8) (b^4)-18 (a^6) (b^6)+12 (a^4) (b^8)-3 (a^2) (b^10)-3 (a^10) (c^2)+9 (a^8) (b^2) (c^2)-6 (a^6) (b^4) (c^2)-6 (a^4) (b^6) (c^2)+9 (a^2) (b^8) (c^2)-3 (b^10) (c^2)+12 (a^8) (c^4)-6 (a^6) (b^2) (c^4)-12 (a^4) (b^4) (c^4)-6 (a^2) (b^6) (c^4)+12 (b^8) (c^4)-18 (a^6) (c^6)-6 (a^4) (b^2) (c^6)-6 (a^2) (b^4) (c^6)-18 (b^6) (c^6)+12 (a^4) (c^8)+9 (a^2) (b^2) (c^8)+12 (b^4) (c^8)-3 (a^2) (c^10)-3 (b^2) (c^10)+32 (a^4) (b^4) (c^4) ((Sin[A])^2) ((Sin[B])^2) ((Sin[C])^2);

f2=Factor[DistanceTrilinears[eqn2]/id]

Simplify[Sqrt[((64 (a^4) (b^4) (c^4) ((Cos[A])^2) ((Cos[B])^2) ((Cos[C])^2))f2+(32 (a^4) (b^4) (c^4) ((Sin[A])^2) ((Sin[B])^2) ((Sin[C])^2)))/f1] R/(4 (a^2) (b^2) (c^2)),TriangleQ[]]

Incident Points

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

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