Nagel Line

Author

Eric W. Weisstein
October 21, 2005

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

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

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

Figure

t=Triangle[{{0.,0.},{3.,-.1},{.8,1.3}}];
Show[Graphics[{
t,
{Red,LongestLine[Coordinates/@{NagelPoint[t],Incenter[t],SpiekerCenter[t],Centroid[t]}]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@{
{NagelPoint[t],"Na"},
{Incenter[t],"I"},
{SpiekerCenter[t],"Sp"},
{Centroid[t],"G"}
}]]
},
AspectRatio->Automatic,PlotRange->All]
]

Equation

Expand[a b c Det[{{α,β,γ},Trilinears[Incenter],Trilinears[Centroid]}]]

Simplify[Coefficient[a b α-a c α-a b β+b c β+a c γ-b c γ,#]&/@{α,β,γ}]

Complement and Anticomplement

Symmetrize[ToSidelengths[Line[NagelLine][[1,1]]]]

Symmetrize[ToSidelengths[TrilinearComplement[NagelLine][[1,1]]]]

Symmetrize[ToSidelengths[Anticomplement[NagelLine][[1,1]]]]

Incident Centers

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

First/@ctrs

Length[ctrs]

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

CenterName/@X/@{34,27,106}

Check

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

Distances

Simplify[Distance[Incenter,SpiekerCenter]/Distance[NagelPoint,SpiekerCenter],TriangleQ[]]

Simplify[Distance[Incenter,Centroid]/Distance[NagelPoint,Centroid],TriangleQ[]]

Radical Line

t=RandomTriangle[20];

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

p=Cartesian[TrilinearLine[NagelLine][[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

CollinearQ[Center/@circs]

Check Radical Axis

Symmetrize[ToSidelengths[First/@(RadicalLine@@@coaxal)][[1]]]

Figure

Name[X[182]]

t=Triangle[{{0,0},{1,0},{-.2,.5}}];
Show[Graphics[{
t,
{Blue,Through[circs[t]]},
{
Red,
l1=NagelLine[t],
{
Dashing[{.015}],
l2=PerpendicularLine[DeLongchampsPoint[t],l1,10]
},
PerpSquare[{l1,l2},.15,4]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@{
{DeLongchampsPoint[t],"(Subscript[X, 20])"},
{Center[YffContactCircle[t]],"Y"}
}
],PointSize[.015],LabelOffset->.1]
}],AspectRatio->Automatic,PlotRange->{{-1.,3.5},{-1.5,1.8}}]