Lucas Tangents Triangle

Author

Eric W. Weisstein
November 8, 2005

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

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

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

Figure

Show[Graphics[{
t,
{Blue,LucasCircles[t]},
{Red,LucasTangentsTriangle[t]},
{Green,LucasCentralTriangle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[LucasTangentsTriangle[t]],{"(Subscript[T, A])","(Subscript[T, B])","(Subscript[T, C])"}}],
Append[#,Blue]&/@Transpose[{Center/@LucasCircles[t],{"(Subscript[L, A])","(Subscript[L, B])","(Subscript[L, C])"}}]
],LabelOffset->.04]
},AspectRatio->Automatic,PlotRange->All]
]

Trilinear Vertex Matrix

TrilinearVertexMatrix[LucasTangentsTriangle]//TraditionalForm

Area

t

{n,d}=Through[{Numerator,Denominator}[Factor[ExpandAll[Together//@ToSidelengths[Area[LucasTangentsTriangle]]]]]]

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

s=(Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)]);
Factor[d/4-(f=Factor[Coefficient[d/4,s]]s)]+f

5 (a-b-c) (a+b-c) (a-b+c) (a+b+c) ((a^2) (b^2)+(a^2) (c^2)+(b^2) (c^2))+(Sqrt[Factor[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)]]) (2 (a^6)-3 (a^4) (b^2)-3 (a^2) (b^4)+2 (b^6)-3 (a^4) (c^2)-14 (a^2) (b^2) (c^2)-3 (b^4) (c^2)-3 (a^2) (c^4)-3 (b^2) (c^4)+2 (c^6))

(-(a^2) (b^2) (c^2) (2 (Sqrt[(-a+b+c) (a+b-c) (a-b+c) (a+b+c)])+ ((a^2)+(b^2)+(c^2))))/(4(-5(Sqrt[(-a+b+c) (a+b-c) (a-b+c) (a+b+c)]) ((a^2) (b^2)+(a^2) (c^2)+(b^2) (c^2))+(2 (a^6)-3 (a^4) (b^2)-3 (a^2) (b^4)+2 (b^6)-3 (a^4) (c^2)-14 (a^2) (b^2) (c^2)-3 (b^4) (c^2)-3 (a^2) (c^4)-3 (b^2) (c^4)+2 (c^6))))

Area[Evaluate[LucasTangentsTriangle[t]]]

(((a^2) (b^2) (c^2) (8Area+((a^2)+(b^2)+(c^2))))/(4(20Area ((a^2) (b^2)+(a^2) (c^2)+(b^2) (c^2))-(2 (a^6)-3 (a^4) (b^2)-3 (a^2) (b^4)+2 (b^6)-3 (a^4) (c^2)-14 (a^2) (b^2) (c^2)-3 (b^4) (c^2)-3 (a^2) (c^4)-3 (b^2) (c^4)+2 (c^6))))//ToSidelengths)/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

Side Lengths

Simplify[SideLengths[LucasTangentsTriangle],TriangleQ[]]

Derivation

circs=Circles[LucasCircles];

t=Triangle[{{0,0},{1,0},{.3,.5}}];

pta=Union[{α,β,γ}/.NSolve[
ToSidelengths[{
Distance[{α,β,γ},Center[circs[[3]]]]==CircleRadius[circs[[3]]],
Distance[{α,β,γ},Center[circs[[2]]]]==CircleRadius[circs[[2]]],
a α+b β+c γ==2Area
}]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]/.Thread[{A,B,C}->Angles[t]]],
{α,β,γ}]][[1]]

Cartesian[pta[t]]

ToAngles[{a(b^2+c^2-a^2)/2,b((a^2+c^2-b^2)/2+2Area),c((a^2+b^2-c^2)/2+2Area)}]

Cartesian[{a(b^2+c^2-a^2)/2,b((a^2+c^2-b^2)/2+2Area),c((a^2+b^2-c^2)/2+2Area)}[t]]

Show[Graphics[{
t,
{Blue,LucasCircles[t]},
PointSize[.02],Cartesian[pta[t]],
{Red,LucasTangentsTriangle[t]}
},AspectRatio->Automatic]
]

pta=Union[{α,β,γ}/.NSolve[
ToSidelengths[{
ExactDistance[{α,β,γ},ExactTrilinears@Center[circs[[3]]]]==CircleRadius[circs[[3]]],
ExactDistance[{α,β,γ},ExactTrilinears@Center[circs[[2]]]]==CircleRadius[circs[[2]]],
a α+b β+c γ==2Area
}]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]/.Thread[{A,B,C}->Angles[t]]],
{α,β,γ}]][[1]]

ctrb=Map[Factor,ToAngles[FullSimplify[Together//@ToSidelengths[ExactTrilinears[Center[circs[[2]]]]],TriangleQ[]]]/.(Sqrt[-((a-b-c) (a+b-c) (a-b+c) (a+b+c))])->4Area,3]

ctrc=Map[Factor,ToAngles[FullSimplify[Together//@ToSidelengths[ExactTrilinears[Center[circs[[3]]]]],TriangleQ[]]]/.(Sqrt[-((a-b-c) (a+b-c) (a-b+c) (a+b+c))])->4Area,3]

Solve[
{
ExactDistance[{α,β,γ},ctrb]^2==CircleRadius[circs[[2]]]^2,
ExactDistance[{α,β,γ},ctrc]^2==CircleRadius[circs[[3]]]^2,
a α+b β+c γ==2Area
},
{α,β,γ}]

Centers

(pts={#,KimberlingLookup[#[LucasTangentsTriangle]]}&/@Take[KimberlingCenters,100]);//Timing

{#[[1]],Name[#[[1]]],X[#[[2,1]]],Name[X[#[[2,1]]]]}&/@Cases[pts,_?(Last[#]=!={}&)]//TableForm