Line at Infinity

Author

Eric W. Weisstein
February 6, 2005

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

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

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

Equation

LineEquation[LineAtInfinity]

Complement and Anticomplement

Symmetrize[ToSidelengths[Line[LineAtInfinity][[1,1]]]]

Symmetrize[ToSidelengths[TrilinearComplement[LineAtInfinity][[1,1]]]]

Symmetrize[ToSidelengths[Anticomplement[LineAtInfinity][[1,1]]]]

Incident Centers

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

First/@ctrs

Length[ctrs]

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

Check

Do[
Print[{i,ctrs[[i]],
FullSimplify[Simplify[DistanceTrilinears[TrigExpand[LineEquation[LineAtInfinity]/.Thread[{α,β,γ}->Trilinears[ctrs[[i]]]]]],TriangleQ[]],TriangleQ[]]//Timing}],
{i,Length[ctrs]}]

Radical Line

samectrs={
{AdamsCircle,ConwayCircle,Incircle},
{BrocardCircle,FirstLemoineCircle},
{Circumcircle,SecondBrocardCircle,SecondDrozFarnyCircle},
{ExtouchTriangleCircumcircle,MandartCircle},
{GallatlyCircle,HalfMosesCircle,MosesCircle},
{NinePointCircle,SteinerCircle},
{PolarCircle,DrozFarnyCircle[1]},
{SecondBrocardCircle,DrozFarnyCircle[2]}
};

Map[CircleCenter,samectrs,{2}]