van Aubel Line

Author

Eric W. Weisstein
April 19, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.6}}];
Show[Graphics[{
t,
{Red,VanAubelLine[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@{
{Orthocenter[t],"H",LabelOffset->{-.04,-.04}},
{SymmedianPoint[t],"K"},
{X[53][t],"(Subscript[X, 53])"}
}
],LabelOffset->.04]
}],AspectRatio->Automatic,PlotRange->{{-.2,1.1},{-.2,.8}}]

Incident Centers

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

First/@ctrs

Length[ctrs]

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

Equation

TrilinearEquation[VanAubelLine]

CyclicSum[a((b^2)-(c^2))(SA^2)α]

Simplify[ToSidelengths[TrigExpand[Divide@@{CyclicSum[a((b^2)-(c^2))(SA^2)α],TrilinearEquation[VanAubelLine]}]],TriangleQ[]]

Simplify[ToAngles[Factor//@%69],Area>0]

Coaxal Circles

t=RandomTriangle[20];

Length[pairs=Subsets[CentralCircles,{2}]]

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

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

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

{Length[circs],circs}

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

(*
t=Triangle[{{0,0},{1,0},{-.35,.5}}];
Show[Graphics[{
t,
{Blue,Through[circs[t]]},
{
Red,
l1=OrthicAxis[t],
{
Dashing[{.015}],
l2=PerpendicularLine[Circumcenter[t],l1,10]
},
PerpSquare[{l1,l2},.15,4]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}]
],PointSize[.015],LabelOffset->.1]
}],AspectRatio->Automatic,PlotRange->{{-2.4,3.5},{-2.3,2}}]
*)

Derivation

lmn=Symmetrize[-ToAngles[Symmetrize[ToSidelengths[TrilinearLine[Line[Orthocenter,SymmedianPoint]][[1]]]]]]

IncidentCenters[KimberlingCenters,lmn]//Timing

Through[{Name,CenterFunction}[X[520]]]