Dou Circle

Author

Eric W. Weisstein
January 17, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.2,.7}}];
colors={Purple,Green,Blue};
circ=DouCircle[t];
lines=Line[Insert[#1,#2,2]]&@@@Transpose[{(Coordinates/@Intersections[circ,#])&/@Sides[t],Vertices[t]}];
Show[Graphics[{
{Dashing[{.015}],Transpose[{colors,Sides[t,2]}]},
{Thickness[.01],t},
{Transpose[{colors,lines}]},
{Red,circ},
Transpose[{colors,PerpSquare[Line/@Partition[Coordinates[#],2,1],.12,4]&/@lines}],
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"},{{LabelOffset->-.1},{},{}}}],
Flatten[Transpose/@Transpose[{Intersections[circ,#]&/@Sides[t],{{"A'","A''"},{"B'","B''"},{"C'","C''"}},
Transpose[Table[colors,{2}]]}],1],
{{X[155][t],"(Subscript[X, 155])",Red}}
],PointSize[.025]]
}],
AspectRatio->Automatic,PlotRange->{{-2,1.5},{-1,2}}]

Dot@@@(Subtract@@@(Partition[Coordinates[#],2,1]&/@lines))//Chop

Circle Function

Factor[CircleFunction[Numerator[Factor[TrilinearEquation[DouCircle]]]][[1]]]

Factor[CircleFunction[DouCircle]]

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

Factor[ToSidelengths[-(a^2 (SA^2 - SB SC) - SA (SB^2 + SC^2)) / (2 SB SC)/(b c)]]

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

Circle Center

CenterName[X[155]]

(center=Simplify[Together//@DistanceTrilinears[CircleCenter[CircleEquation[DouCircle]]],TriangleQ[]])//Timing

Symmetrize/@center

Symmetrize/@LawOfCosines[%]

CircleCenter[DouCircle]

Circle Radius

CircleRadius[DouCircle]//TraditionalForm

Simplify[Together//@CircleRadius[CircleEquation[DouCircle]],TriangleQ[]]

t=Triangle[{{0,0},{1,0},{.3,.7}}];

DouCircle[t]

ToSidelengths[(√((a^18)-7 (a^16) (b^2)+20 (a^14) (b^4)-28 (a^12) (b^6)+14 (a^10) (b^8)+14 (a^8) (b^10)-28 (a^6) (b^12)+20 (a^4) (b^14)-7 (a^2) (b^16)+(b^18)-7 (a^16) (c^2)+34 (a^14) (b^2) (c^2)-64 (a^12) (b^4) (c^2)+62 (a^10) (b^6) (c^2)-50 (a^8) (b^8) (c^2)+62 (a^6) (b^10) (c^2)-64 (a^4) (b^12) (c^2)+34 (a^2) (b^14) (c^2)-7 (b^16) (c^2)+20 (a^14) (c^4)-64 (a^12) (b^2) (c^4)+72 (a^10) (b^4) (c^4)-28 (a^8) (b^6) (c^4)-28 (a^6) (b^8) (c^4)+72 (a^4) (b^10) (c^4)-64 (a^2) (b^12) (c^4)+20 (b^14) (c^4)-28 (a^12) (c^6)+62 (a^10) (b^2) (c^6)-28 (a^8) (b^4) (c^6)+20 (a^6) (b^6) (c^6)-28 (a^4) (b^8) (c^6)+62 (a^2) (b^10) (c^6)-28 (b^12) (c^6)+14 (a^10) (c^8)-50 (a^8) (b^2) (c^8)-28 (a^6) (b^4) (c^8)-28 (a^4) (b^6) (c^8)-50 (a^2) (b^8) (c^8)+14 (b^10) (c^8)+14 (a^8) (c^10)+62 (a^6) (b^2) (c^10)+72 (a^4) (b^4) (c^10)+62 (a^2) (b^6) (c^10)+14 (b^8) (c^10)-28 (a^6) (c^12)-64 (a^4) (b^2) (c^12)-64 (a^2) (b^4) (c^12)-28 (b^6) (c^12)+20 (a^4) (c^14)+34 (a^2) (b^2) (c^14)+20 (b^4) (c^14)-7 (a^2) (c^16)-7 (b^2) (c^16)+(c^18)) Abs[Sec[A] Sec[B] Sec[C]])/(8 (Sqrt[2]) (a^2) (b^2) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]))]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

Incident Centers

IncidentCenters[KimberlingCenters,DouCircle]//Timing

Derivation

LineEquation[{{0,1,0},{0,0,1}}]

CenterFunction[X[155]]

Symmetrize[Simplify[Together//@DistanceTrilinears[CenterFunction[X[155]]],TriangleQ[]]]

Simplify[Together//@DistanceTrilinears[CircleEquation[X[155],r]],TriangleQ[]]

eqn=Numerator[%];

Solve[{eqn==0,α==0,2Area[]==(a α+b β+c γ)},{α,β,γ}]

soln=Simplify[%,TriangleQ[]]

gamma=((a^14) b (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-4 (a^12) (b^3) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+5 (a^10) (b^5) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-5 (a^6) (b^9) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+4 (a^4) (b^11) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-(a^2) (b^13) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+6 (a^10) (b^3) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-12 (a^8) (b^5) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+8 (a^6) (b^7) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-4 (a^4) (b^9) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+2 (a^2) (b^11) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-7 (a^10) b (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+4 (a^8) (b^3) (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+10 (a^6) (b^5) (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-4 (a^4) (b^7) (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-3 (a^2) (b^9) (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+8 (a^8) b (c^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-16 (a^6) (b^3) (c^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-4 (a^4) (b^5) (c^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+4 (a^2) (b^7) (c^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+3 (a^6) b (c^8) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+16 (a^4) (b^3) (c^8) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+(a^2) (b^5) (c^8) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-8 (a^4) b (c^10) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])-6 (a^2) (b^3) (c^10) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+3 (a^2) b (c^12) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])+√(-(a^2) (b^2) (((-(a^2)+(b^2)+(c^2)))^2) ((((a^6)-(a^2) ((((b^2)-(c^2)))^2)-(a^4) ((b^2)+(c^2))+((((b^2)-(c^2)))^2) ((b^2)+(c^2))))^2) ((a^14)+4 ((((b^2)-(c^2)))^6) (r^2)+(a^2) ((((b^2)-(c^2)))^4) ((b^2)+(c^2)) ((b^2)+(c^2)-8 (r^2))+(a^12) (-6 (b^2)-6 (c^2)+4 (r^2))+(a^10) (15 (b^4)+15 (c^4)-8 (c^2) (r^2)+2 (b^2) (7 (c^2)-4 (r^2)))-2 (a^4) ((((b^2)-(c^2)))^2) (3 (b^6)+3 (c^6)+2 (c^4) (r^2)+(b^4) ((c^2)+2 (r^2))+(b^2) ((c^4)-4 (c^2) (r^2)))-4 (a^8) (5 (b^6)+(b^4) ((c^2)+(r^2))+(c^4) (5 (c^2)+(r^2))+(b^2) ((c^4)-2 (c^2) (r^2)))+(a^6) (15 (b^8)+15 (c^8)+16 (c^6) (r^2)-4 (b^6) (3 (c^2)-4 (r^2))+2 (b^4) (5 (c^4)-8 (c^2) (r^2))-4 (b^2) (3 (c^6)+4 (c^4) (r^2))))))/(4 (a^2) b c ((((a^6)-(a^2) ((((b^2)-(c^2)))^2)-(a^4) ((b^2)+(c^2))+((((b^2)-(c^2)))^2) ((b^2)+(c^2))))^2));

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

Simplify[(√(-(a^2) (b^2) (((-(a^2)+(b^2)+(c^2)))^2) ((((a^6)-(a^2) ((((b^2)-(c^2)))^2)-(a^4) ((b^2)+(c^2))+((((b^2)-(c^2)))^2) ((b^2)+(c^2))))^2) ((a^14)+4 ((((b^2)-(c^2)))^6) (r^2)+(a^2) ((((b^2)-(c^2)))^4) ((b^2)+(c^2)) ((b^2)+(c^2)-8 (r^2))+(a^12) (-6 (b^2)-6 (c^2)+4 (r^2))+(a^10) (15 (b^4)+15 (c^4)-8 (c^2) (r^2)+2 (b^2) (7 (c^2)-4 (r^2)))-2 (a^4) ((((b^2)-(c^2)))^2) (3 (b^6)+3 (c^6)+2 (c^4) (r^2)+(b^4) ((c^2)+2 (r^2))+(b^2) ((c^4)-4 (c^2) (r^2)))-4 (a^8) (5 (b^6)+(b^4) ((c^2)+(r^2))+(c^4) (5 (c^2)+(r^2))+(b^2) ((c^4)-2 (c^2) (r^2)))+(a^6) (15 (b^8)+15 (c^8)+16 (c^6) (r^2)-4 (b^6) (3 (c^2)-4 (r^2))+2 (b^4) (5 (c^4)-8 (c^2) (r^2))-4 (b^2) (3 (c^6)+4 (c^4) (r^2))))))/(4 (a^2) b c ((((a^2)-(b^2)-(c^2)))^2) ((((a^2)+(b^2)-(c^2)))^2) ((((a^2)-(b^2)+(c^2)))^2)),TriangleQ[]&&r>0]

(√Factor[(-(a^14)-4 ((((b^2)-(c^2)))^6) (r^2)-(a^2) ((((b^2)-(c^2)))^4) ((b^2)+(c^2)) ((b^2)+(c^2)-8 (r^2))+(a^12) (6 (b^2)+6 (c^2)-4 (r^2))+(a^10) (-15 (b^4)-15 (c^4)+8 (c^2) (r^2)+(b^2) (-14 (c^2)+8 (r^2)))+2 (a^4) ((((b^2)-(c^2)))^2) (3 (b^6)+3 (c^6)+2 (c^4) (r^2)+(b^4) ((c^2)+2 (r^2))+(b^2) ((c^4)-4 (c^2) (r^2)))+4 (a^8) (5 (b^6)+(b^4) ((c^2)+(r^2))+(c^4) (5 (c^2)+(r^2))+(b^2) ((c^4)-2 (c^2) (r^2)))+(a^6) (-15 (b^8)-15 (c^8)-16 (c^6) (r^2)+4 (b^6) (3 (c^2)-4 (r^2))-2 (b^4) (5 (c^4)-8 (c^2) (r^2))+4 (b^2) (3 (c^6)+4 (c^4) (r^2)))) ]Abs[-(a^2)+(b^2)+(c^2)] Abs[Factor[(a^6)-(a^2) ((((b^2)-(c^2)))^2)-(a^4) ((b^2)+(c^2))+((((b^2)-(c^2)))^2) ((b^2)+(c^2))]])/(4 a c ((((a^2)+(b^2)-(c^2)))^2) ((((a^2)-(b^2)+(c^2)))^2) (((-(a^2)+(b^2)+(c^2)))^2))

soln={α,β,γ}/.%11;

lines=LineEquation[{{1,0,0},#}]&/@soln;

?CoefficientList

{l,m,n}=Coefficient[lines[[1]],#]&/@{α,β,γ};

{ll,mm,nn}=Coefficient[lines[[2]],#]&/@{α,β,γ};

eqn=l ll+m mm+n nn-(m nn+mm n)Cos[A]-(n ll+nn l)Cos[B]-(l mm+ll m)Cos[C]

soln=Solve[eqn==0,r]

rsoln=(r/.soln)[[2]]

Circle Radius

Together//@DistanceTrilinears[rsoln]

Factor[((-(a^20)+7 (a^18) (b^2)-20 (a^16) (b^4)+28 (a^14) (b^6)-14 (a^12) (b^8)-14 (a^10) (b^10)+28 (a^8) (b^12)-20 (a^6) (b^14)+7 (a^4) (b^16)-(a^2) (b^18)+7 (a^18) (c^2)-34 (a^16) (b^2) (c^2)+64 (a^14) (b^4) (c^2)-62 (a^12) (b^6) (c^2)+50 (a^10) (b^8) (c^2)-62 (a^8) (b^10) (c^2)+64 (a^6) (b^12) (c^2)-34 (a^4) (b^14) (c^2)+7 (a^2) (b^16) (c^2)-20 (a^16) (c^4)+64 (a^14) (b^2) (c^4)-72 (a^12) (b^4) (c^4)+28 (a^10) (b^6) (c^4)+28 (a^8) (b^8) (c^4)-72 (a^6) (b^10) (c^4)+64 (a^4) (b^12) (c^4)-20 (a^2) (b^14) (c^4)+28 (a^14) (c^6)-62 (a^12) (b^2) (c^6)+28 (a^10) (b^4) (c^6)-20 (a^8) (b^6) (c^6)+28 (a^6) (b^8) (c^6)-62 (a^4) (b^10) (c^6)+28 (a^2) (b^12) (c^6)-14 (a^12) (c^8)+50 (a^10) (b^2) (c^8)+28 (a^8) (b^4) (c^8)+28 (a^6) (b^6) (c^8)+50 (a^4) (b^8) (c^8)-14 (a^2) (b^10) (c^8)-14 (a^10) (c^10)-62 (a^8) (b^2) (c^10)-72 (a^6) (b^4) (c^10)-62 (a^4) (b^6) (c^10)-14 (a^2) (b^8) (c^10)+28 (a^8) (c^12)+64 (a^6) (b^2) (c^12)+64 (a^4) (b^4) (c^12)+28 (a^2) (b^6) (c^12)-20 (a^6) (c^14)-34 (a^4) (b^2) (c^14)-20 (a^2) (b^4) (c^14)+7 (a^4) (c^16)+7 (a^2) (b^2) (c^16)-(a^2) (c^18)))/( 2((a^18)-4 (a^16) (b^2)+4 (a^14) (b^4)+4 (a^12) (b^6)-10 (a^10) (b^8)+4 (a^8) (b^10)+4 (a^6) (b^12)-4 (a^4) (b^14)+(a^2) (b^16)-4 (a^16) (c^2)+12 (a^14) (b^2) (c^2)-12 (a^12) (b^4) (c^2)+4 (a^10) (b^6) (c^2)+4 (a^8) (b^8) (c^2)-12 (a^6) (b^10) (c^2)+12 (a^4) (b^12) (c^2)-4 (a^2) (b^14) (c^2)+4 (a^14) (c^4)-12 (a^12) (b^2) (c^4)+12 (a^10) (b^4) (c^4)-8 (a^8) (b^6) (c^4)+12 (a^6) (b^8) (c^4)-12 (a^4) (b^10) (c^4)+4 (a^2) (b^12) (c^4)+4 (a^12) (c^6)+4 (a^10) (b^2) (c^6)-8 (a^8) (b^4) (c^6)-8 (a^6) (b^6) (c^6)+4 (a^4) (b^8) (c^6)+4 (a^2) (b^10) (c^6)-10 (a^10) (c^8)+4 (a^8) (b^2) (c^8)+12 (a^6) (b^4) (c^8)+4 (a^4) (b^6) (c^8)-10 (a^2) (b^8) (c^8)+4 (a^8) (c^10)-12 (a^6) (b^2) (c^10)-12 (a^4) (b^4) (c^10)+4 (a^2) (b^6) (c^10)+4 (a^6) (c^12)+12 (a^4) (b^2) (c^12)+4 (a^2) (b^4) (c^12)-4 (a^4) (c^14)-4 (a^2) (b^2) (c^14)+(a^2) (c^16)))]

Simplify[Sqrt[-2 (a-b-c) (a+b-c) (a-b+c) (a+b+c) ((((a^2)-(b^2)-(c^2)))^2) ((((a^2)+(b^2)-(c^2)))^2) ((((a^2)-(b^2)+(c^2)))^2)],TriangleQ[]]

1/Simplify[Sqrt[-2 (a-b-c) (a+b-c) (a-b+c) (a+b+c) ((((a^2)-(b^2)-(c^2)))^2) ((((a^2)+(b^2)-(c^2)))^2) ((((a^2)-(b^2)+(c^2)))^2)],TriangleQ[]](√((a^18)-7 (a^16) (b^2)+20 (a^14) (b^4)-28 (a^12) (b^6)+14 (a^10) (b^8)+14 (a^8) (b^10)-28 (a^6) (b^12)+20 (a^4) (b^14)-7 (a^2) (b^16)+(b^18)-7 (a^16) (c^2)+34 (a^14) (b^2) (c^2)-64 (a^12) (b^4) (c^2)+62 (a^10) (b^6) (c^2)-50 (a^8) (b^8) (c^2)+62 (a^6) (b^10) (c^2)-64 (a^4) (b^12) (c^2)+34 (a^2) (b^14) (c^2)-7 (b^16) (c^2)+20 (a^14) (c^4)-64 (a^12) (b^2) (c^4)+72 (a^10) (b^4) (c^4)-28 (a^8) (b^6) (c^4)-28 (a^6) (b^8) (c^4)+72 (a^4) (b^10) (c^4)-64 (a^2) (b^12) (c^4)+20 (b^14) (c^4)-28 (a^12) (c^6)+62 (a^10) (b^2) (c^6)-28 (a^8) (b^4) (c^6)+20 (a^6) (b^6) (c^6)-28 (a^4) (b^8) (c^6)+62 (a^2) (b^10) (c^6)-28 (b^12) (c^6)+14 (a^10) (c^8)-50 (a^8) (b^2) (c^8)-28 (a^6) (b^4) (c^8)-28 (a^4) (b^6) (c^8)-50 (a^2) (b^8) (c^8)+14 (b^10) (c^8)+14 (a^8) (c^10)+62 (a^6) (b^2) (c^10)+72 (a^4) (b^4) (c^10)+62 (a^2) (b^6) (c^10)+14 (b^8) (c^10)-28 (a^6) (c^12)-64 (a^4) (b^2) (c^12)-64 (a^2) (b^4) (c^12)-28 (b^6) (c^12)+20 (a^4) (c^14)+34 (a^2) (b^2) (c^14)+20 (b^4) (c^14)-7 (a^2) (c^16)-7 (b^2) (c^16)+(c^18)))

(Sqrt[2])DistanceTrilinears[((8Cos[A]Cos[B]Cos[C](a b c)^3)/(Circumradius[]))]

((Sqrt[f])/((8Cos[A]Cos[B]Cos[C](a b c)^3)/R))

((((a^18)-7 (a^16) (b^2)+20 (a^14) (b^4)-28 (a^12) (b^6)+14 (a^10) (b^8)+14 (a^8) (b^10)-28 (a^6) (b^12)+20 (a^4) (b^14)-7 (a^2) (b^16)+(b^18)-7 (a^16) (c^2)+34 (a^14) (b^2) (c^2)-64 (a^12) (b^4) (c^2)+62 (a^10) (b^6) (c^2)-50 (a^8) (b^8) (c^2)+62 (a^6) (b^10) (c^2)-64 (a^4) (b^12) (c^2)+34 (a^2) (b^14) (c^2)-7 (b^16) (c^2)+20 (a^14) (c^4)-64 (a^12) (b^2) (c^4)+72 (a^10) (b^4) (c^4)-28 (a^8) (b^6) (c^4)-28 (a^6) (b^8) (c^4)+72 (a^4) (b^10) (c^4)-64 (a^2) (b^12) (c^4)+20 (b^14) (c^4)-28 (a^12) (c^6)+62 (a^10) (b^2) (c^6)-28 (a^8) (b^4) (c^6)+20 (a^6) (b^6) (c^6)-28 (a^4) (b^8) (c^6)+62 (a^2) (b^10) (c^6)-28 (b^12) (c^6)+14 (a^10) (c^8)-50 (a^8) (b^2) (c^8)-28 (a^6) (b^4) (c^8)-28 (a^4) (b^6) (c^8)-50 (a^2) (b^8) (c^8)+14 (b^10) (c^8)+14 (a^8) (c^10)+62 (a^6) (b^2) (c^10)+72 (a^4) (b^4) (c^10)+62 (a^2) (b^6) (c^10)+14 (b^8) (c^10)-28 (a^6) (c^12)-64 (a^4) (b^2) (c^12)-64 (a^2) (b^4) (c^12)-28 (b^6) (c^12)+20 (a^4) (c^14)+34 (a^2) (b^2) (c^14)+20 (b^4) (c^14)-7 (a^2) (c^16)-7 (b^2) (c^16)+(c^18)))^(1/2))(( Circumradius[] Abs[Sec[A] Sec[B] Sec[C]])/(8 (a^3) (b^3) (c^3)(Sqrt[2])))//InputForm