Lemoine Axis

Author

Eric W. Weisstein
October 21, 2005

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

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

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

Equation

LineEquation[LemoineAxis]

Incident Centers

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

First/@ctrs

Length[ctrs]

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

Check

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

Radical Line

t=RandomTriangle[20];

Length[pairs=Subsets[Select[CentralCircles,(CircleFunction[#]=!=Null&)],{2}]]

p=Cartesian[TrilinearLine[LemoineAxis][[1]][t]]

(coaxal=Select[pairs,Cartesian[First[RadicalLine@@#][t]]==p&])//Timing

Coaxal Systems

Subsets[circs=Union[Flatten[coaxal]],{2}]==coaxal

{Length[circs],circs}

Check Collinearity

Refine[Together//@ToSidelengths[CollinearQ[Center/@circs]/.cos_Cos:>TrigExpand[cos]],TriangleQ[]]

Check Radical Axis

First/@Symmetrize/@ToAngles/@Symmetrize/@ToSidelengths[First/@(RadicalLine@@@coaxal)]

Figure

t=Triangle[{{0,0},{1,0},{.1,.5}}];
Show[Graphics[{
t,
{Blue,Through[circs[t]]},
{
Red,
l1=LemoineAxis[t],
{
Dashing[{.015}],
l2=PerpendicularLine[Circumcenter[t],l1,10]
},
PerpSquare[{l1,l2},.05,4]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@{
{X[182][t],"(Subscript[X, 182])",LabelOffset->{-.17,.02}},
{Circumcenter[t],"O"},
{Center[LucasInnerCircle[t]],"L",LabelOffset->{0,-.08}},
{X[1158][t],"(Subscript[X, 1158])",LabelOffset->{0,-.08}}
}
],PointSize[.02],LabelOffset->.05]
}],AspectRatio->Automatic,PlotRange->{{-.6,1.5},{-.5,1}}
]