Sine-Triple-Angle Circle

Author

Eric W. Weisstein
November 28, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/Sine-Triple-AngleCircle.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Sine-Triple-AngleCircle.html.

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

Code

SineTripleAngleTriangles[t_Triangle]:=Module[{l=Transpose[Map[Coordinates,Intersections[SineTripleAngleCircle[t],#]&/@Sides[t],{2}]],
ts,
ang1,ang2
},
ts=Triangle/@Flatten[Outer[List,Sequence@@Transpose[l],1],2];
ang1=Map[Angle,Join@@@Transpose[{List/@Vertices[t],Coordinates/@Sides[#]}]&/@ts,{2}];
ang2=Map[Angle,Join@@@Transpose[{List/@Vertices[t],Reverse/@Coordinates/@Sides[#]}]&/@ts,{2}];
ts[[Flatten[Position[#,_?(#==Angles[t]&)]&/@{ang1,ang2}]]]
]

Figure

t=Triangle[{{0,0},{1,0},{.44,.6}}];
sta=SineTripleAngleTriangles[t];
vs1=Join@@@Transpose[{List/@Vertices[t],Reverse/@Coordinates/@Sides[sta[[2]]]}];
vs2=Join@@@Transpose[{List/@Vertices[t],Coordinates/@Sides[sta[[1]]]}];
Show[Graphics[{
t,
{Red,circ=SineTripleAngleCircle[t]},
Transpose[{{Blue,Purple},sta}],
VertexArcs[t,.07],
MapIndexed[EqualAngleTicks[#1,Sequence@@#2,.07,.015,.2]&,NestList[RotateLeft,RotateRight[Vertices[t]],2]],
{Purple,
VertexArc[#,.07]&/@vs1,
MapIndexed[EqualAngleTicks[#1,Sequence@@#2,.07,.015,.2]&,vs1]
},
{Blue,
VertexArc[#,.07]&/@vs2,
MapIndexed[EqualAngleTicks[#1,Sequence@@#2,.07,.015,.2]&,vs2]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Flatten[Transpose/@Transpose[{VertexPoints/@sta,{{"(Subscript[A, 1])","(Subscript[B, 1])","(Subscript[C, 1])"},{"(Subscript[A, 2])","(Subscript[B, 2])","(Subscript[C, 2])"}},{Table[Blue,{3}],Table[Purple,{3}]}}],1],
{
{Center[circ],"O",Red}
}
],LabelOffset->-.04,PointSize[.025]]
}],AspectRatio->Automatic]

Verify angles

{aa,bb,cc}=Vertices[t];
{a2,b2,c2}=Vertices[sta[[1]]];
{a1,b1,c1}=Vertices[sta[[2]]];

{Angles[Triangle[{aa,b1,c1}]],Angles[Triangle[{bb,c1,a1}]],Angles[Triangle[{cc,b1,a1}]]}

{Angles[Triangle[{aa,b2,c2}]],Angles[Triangle[{bb,c2,a2}]],Angles[Triangle[{cc,b2,a2}]]}

Verify triple-angle

#/Min[#]&[Distance@@@Transpose[Vertices/@sta]]

#/Min[Abs[#]]&[Sin/@(3Angles[t])]

Circle Function

CircleFunction[SineTripleAngleCircle]

Derivation

tri=FullSimplify[Together//@DistanceTrilinears[TrigExpand[Trilinears[X[49]]]]]

Factor[FullSimplify[Together//@DistanceTrilinears[CircleFunction[CircleEquation[tri,Circumradius[]/(1+8 Cos[A]Cos[B]Cos[C])]]],TriangleQ[]]]

Simplification

CircleFunction[SineTripleAngleCircle]

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

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

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

Together//@DistanceTrilinears[-(a^2) (b^2) (c^2) (1+8 Cos[A] Cos[B] Cos[C])]

f=-(b c (-a b+(b^2)-(c^2)) (a b+(b^2)-(c^2)) ((b^2)-a c-(c^2)) ((b^2)+a c-(c^2)) (-(a^2)+(b^2)+(c^2)))/((a^2) (b^2) (c^2) (1+8 Cos[A] Cos[B] Cos[C]))^2

LawOfCosines[f]

Kimberling Center

IncidentCenters[KimberlingCenters,CyclicTrilinears[-(b c (-a b+(b^2)-(c^2)) (a b+(b^2)-(c^2)) ((b^2)-a c-(c^2)) ((b^2)+a c-(c^2)) (-(a^2)+(b^2)+(c^2)))/((((a^6)-(a^4) (b^2)-(a^2) (b^4)+(b^6)-(a^4) (c^2)+(a^2) (b^2) (c^2)-(b^4) (c^2)-(a^2) (c^4)-(b^2) (c^4)+(c^6)))^2)]]

BisymmetricQ[-(b c (-a b+(b^2)-(c^2)) (a b+(b^2)-(c^2)) ((b^2)-a c-(c^2)) ((b^2)+a c-(c^2)) (-(a^2)+(b^2)+(c^2)))/((((a^6)-(a^4) (b^2)-(a^2) (b^4)+(b^6)-(a^4) (c^2)+(a^2) (b^2) (c^2)-(b^4) (c^2)-(a^2) (c^4)-(b^2) (c^4)+(c^6)))^2)]

Center

CircleFunction[SineTripleAngleCircle]

InputForm[%]

IncidentCenters[KimberlingCenters,CyclicTrilinears[CircleFunction[SineTripleAngleCircle]]]//Timing

Circle Radius

r=CircleRadius[SineTripleAngleCircle]/.Cos[x_]:>TrigExpand[Cos[x]]

Simplify[DistanceTrilinears[r],TriangleQ[]]

Refine[%,TriangleQ[]]

Cancel[((a^3) (b^3) (c^3))/((Sqrt[Factor[-(a^4)-((((b^2)-(c^2)))^2)+2 (a^2) ((b^2)+(c^2))]]) Abs[Factor[(a^6)-(a^4) ((b^2)+(c^2))+((((b^2)-(c^2)))^2) ((b^2)+(c^2))-(a^2) ((b^4)-(b^2) (c^2)+(c^4))]])(R/(Circumradius[]))]

Simplify[LawOfCosines[%35],TriangleQ[]]

Table[(1+8Cos[A] Cos[B] Cos[C])>0 /.Thread[{A,B,C}->Angles[Evaluate[Triangle[Table[Random[Real,{0,1},20],{3},{2}]]]]],{20}]

Table[(a^6)-(a^4) ((b^2)+(c^2))+((((b^2)-(c^2)))^2) ((b^2)+(c^2))-(a^2) ((b^4)-(b^2) (c^2)+(c^4))>=0 /.Thread[{a,b,c}->SideLengths[Evaluate[Triangle[Table[Random[Real,{0,1},20],{3},{2}]]]]],{20}]

DistanceTrilinears[1+8Cos[A]Cos[B]Cos[C]]

Together[Factor[%36]]

LawOfCosines[f_]:=f/.Thread[Rule@@(CyclicTrilinears/@{a^2+b^2-c^2,2a b Cos[C]})]/.Thread[Rule@@(CyclicTrilinears/@{-a^2-b^2+c^2,-2a b Cos[C]})]/.a^6-a^4*b^2-a^2*b^4+b^6-a^4*c^2+a^2*b^2*c^2-b^4*c^2-a^2*c^4-b^2*c^4+c^6->-a^2*b^2*c^2*(1+8Cos[A]*Cos[B]*Cos[C])/.-a^6+a^4*b^2+a^2*b^4-b^6+a^4*c^2-a^2*b^2*c^2+b^4*c^2+a^2*c^4+b^2*c^4-c^6->a^2*b^2*c^2*(1+8Cos[A]*Cos[B]*Cos[C])

Incident Points

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

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