M'Cay Cubic

Author

Eric W. Weisstein
December 24, 2004

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

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

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

Pivot Point

CubicFunction[MCayCubic]

CenterFunction[Circumcenter]

Curve

CubicEquation[MCayCubic]

Incident Points

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

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

Check

Not in Kimberling's list

trilinears=Join[Trilinears/@ctrs,TrilinearVertexMatrix[ExcentralTriangle]];

Simplify[DistanceTrilinears[TrigExpand[CubicEquation[MCayCubic]/.Thread[{α,β,γ}->#]]],
TriangleQ[]]&/@trilinears//Timing

Graphic

Solve[cubic==0/.γ->1,β]

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

t=Triangle[{{0,0},{1,0},{.3,.6}}];
Show[Graphics[{
t,
{Purple,
Branches[ParametricPlot[Evaluate[TrilinearToCartesian[t,{α,(-Cos[B]+(α^2) Cos[B]-√((((Cos[B]-(α^2) Cos[B]))^2)-4 (α Cos[A]-Cos[C]) (-α Cos[A]+(α^2) Cos[C])))/(2 (α Cos[A]-Cos[C])),1}/.Thread[{A,B,C}->Angles[t]]]],{α,-30,30},PlotRange->All,PlotPoints->100,
DisplayFunction->Identity][[1]]],
Branches[ParametricPlot[Evaluate[TrilinearToCartesian[t,{α,(-Cos[B]+(α^2) Cos[B]+√((((Cos[B]-(α^2) Cos[B]))^2)-4 (α Cos[A]-Cos[C]) (-α Cos[A]+(α^2) Cos[C])))/(2 (α Cos[A]-Cos[C])),1}/.Thread[{A,B,C}->Angles[t]]]],{α,-30,30},PlotRange->All,PlotPoints->100,
DisplayFunction->Identity][[1]]]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{Excenters[t],{"(Subscript[J, A])","(Subscript[J, B])","(Subscript[J, C])"}}],
Append[#,Red]&/@{
{Incenter[t],"I"},
{Circumcenter[t],"O"},
{Orthocenter[t],"H"},
{X[1075][t],"(Subscript[X, 1075])"}
}
],LabelOffset->.07,PointSize[.03]]
}],AspectRatio->Automatic,PlotRange->{{-.4,1.6},{-1.1,1.}},ImageSize->200]