MacBeath Circumconic

Author

Eric W. Weisstein
December 18, 2004

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

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

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

Equation

Trilinear Equation

eqn=TrilinearEquation[MacBeathCircumconic]

Center

Numerator[Factor[TrilinearEquation[MacBeathCircumconic]]]

-Symmetrize/@ToSidelengths[Center[MacBeathCircumconic]]

IncidentCenters[Take[KimberlingCenters,10],%]//Timing

CenterName[X[6]]

Figure

FullSimplify[Solve[eqn==0/.γ->1,β],TriangleQ[]]

Off[ParametricPlot::"pptr"];
t=Triangle[{{0,0},{1,0},{.3,.8}}];
bp=Cartesian[BrianchonPoint[MacBeathInconic][t]];
Show[Graphics[{
t,
(*Cartesian[Triangle[{{0,(c^2) Cos[C] Sec[B],(b^2)},{(c^2),0,(a^2) Cos[A] Sec[C]},{(b^2) Cos[B] Sec[A],(a^2),0}}][t]],*)
{Red,
ParametricPlot[Evaluate[TrilinearToCartesian[t,{α,-((α Cos[B])/(Cos[A]+α Cos[C])),1}/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]]/.Thread[{A,B,C}->Angles[Evaluate[t]]]],{α,-100,100},PlotRange->All,PlotPoints->100,
DisplayFunction->Identity][[1,1,1]]&/@{-1,1}
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{
{SymmedianPoint[t],"K",{Red,LabelOffset->{.03,.05}}},
{X[110][t],"(Subscript[X, 110])",Red},
{X[648][t],"(Subscript[X, 648])",Red}
}
],
{PointSize[.025],LabelOffset->-.04}
]
}],
PlotRange->{{-.4,1.1},{-.4,.9}},AspectRatio->Automatic]

Incident Centers

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

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