Lucas Cubic

Author

Eric W. Weisstein
January 10, 2005

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

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

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

Pivot Point

CenterFunction[X[69]]

Curve

TrilinearEquation[LucasCubic]

Simplify[Numerator[Together[TrilinearEquation[LucasCubic]/.Thread[{α,β,γ}->IsogonalConjugate[{α,β,γ}]]]]]

Isotomic Conjugate

Simplify[-(a^2) (b^2) (c^2) (α^2) (β^2) (γ^2) (TrilinearEquation[LucasCubic]/.Thread[{α,β,γ}->IsotomicConjugate[{α,β,γ}]])]

Cyclocevian Conjugate

Factor[ToSidelengths[(TrilinearEquation[LucasCubic]/.Thread[{α,β,γ}->CyclocevianConjugate[{α,β,γ}]])]]

eqn=((a^4) b (α^2) β-(a^2) (b^3) (α^2) β+(a^2) b (c^2) (α^2) β+(a^3) (b^2) α (β^2)-a (b^4) α (β^2)-a (b^2) (c^2) α (β^2)-(a^4) c (α^2) γ-(a^2) (b^2) c (α^2) γ+(a^2) (c^3) (α^2) γ+(a^2) (b^2) c (β^2) γ+(b^4) c (β^2) γ-(b^2) (c^3) (β^2) γ-(a^3) (c^2) α (γ^2)+a (b^2) (c^2) α (γ^2)+a (c^4) α (γ^2)-(a^2) b (c^2) β (γ^2)+(b^3) (c^2) β (γ^2)-b (c^4) β (γ^2));

Simplify[eqn-{α,β,γ}.(Coefficient[eqn,#]&/@{α,β,γ})]

Simplify[(1/(2 a b c)){α,β,γ}.ToAngles[Factor[Coefficient[eqn,#]]&/@{α,β,γ}]]

Incident Points

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

Length[ctrs]

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

Check

Graphic

t=Triangle[{{0,0},{1,0},{.3,.6}}];
beta=β/.Solve[TrilinearEquation[LucasCubic]==0/.γ->1,β]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]/.Thread[{A,B,C}->Angles[t]];

Branches[g_]:=Module[{data=Flatten[g][[1,1]]},
split=Split[data,Sqrt[#.#]&[Subtract@@@#]<10&];
{Line/@(Drop[#,-1]&/@split)
}
]

Show[Graphics[{
t,
{Red,
Branches[
ParametricPlot[
Evaluate[TrilinearToCartesian[t,{α,#,1}]],{α,-30,30},PlotRange->All,PlotPoints->200,
DisplayFunction->Identity][[1]]
]&/@beta
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@{
{X[2][t],"G"},
{X[4][t],"H"},
{X[7][t],"Ge"},
{X[8][t],"Na"},
{X[20][t],"L"},
{X[69][t],"(Subscript[X, 69])"},
{X[189][t],"(Subscript[X, 189])"},
{X[253][t],"(Subscript[X, 253])"},
{X[329][t],"(Subscript[X, 329])"},
{X[1032][t],"(Subscript[X, 1032])"},
{X[1034][t],"(Subscript[X, 1034])"}
}
],LabelOffset->.06,PointSize[.02]]
}],AspectRatio->Automatic,PlotRange->{{-.5,1.5},{-.5,1.25}},ImageSize->300]