Orthocubic

Author

Eric W. Weisstein
October 19, 2004

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

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

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

Pivot Point

CubicFunction[Orthocubic]

CenterFunction[Orthocenter]

Curve

CubicEquation[Orthocubic]

Incident Points

(ctrs=IncidentCenters[KimberlingCenters,CubicEquation[Orthocubic]])//Timing

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

Check

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

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

Graphic

FullSimplify[Solve[cubic==0/.β->1,γ]]

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,
{Red,
Branches[ParametricPlot[Evaluate[TrilinearToCartesian[t,{α,1,-(1/(2 (Cos[A]-α Cos[B])))(((-1+(α^2)) Cos[A] Cos[B]+√((((-1+(α^2)))^2) ((Cos[A])^2) ((Cos[B])^2)+4 α (α Cos[A]-Cos[B]) (Cos[A]-α Cos[B]) ((Cos[C])^2))) Sec[C])}/.Thread[{A,B,C}->Angles[t]]]],{α,-30,30},PlotRange->All,PlotPoints->100,
DisplayFunction->Identity][[1]]],
Branches[ParametricPlot[Evaluate[TrilinearToCartesian[t,{α,1,(1/(2 (Cos[A]-α Cos[B])))((-(-1+(α^2)) Cos[A] Cos[B]+√((((-1+(α^2)))^2) ((Cos[A])^2) ((Cos[B])^2)+4 α (α Cos[A]-Cos[B]) (Cos[A]-α Cos[B]) ((Cos[C])^2))) Sec[C])}/.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"},
Sequence@@({X[#][t],Subscript["X",ToString[#]]}&/@{46,90,155,254,371,372,485,486,487,488})
}
],LabelOffset->.07,PointSize[.02]]
}],AspectRatio->Automatic,PlotRange->{{-1,1.6},{-1.1,1.}},ImageSize->300]