Antiorthic Axis

Author

Eric W. Weisstein
October 21, 2005

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

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

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

Equation

LineEquation[AntiorthicAxis]

Points

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

First/@ctrs

Length[ctrs]

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

Check

Simplify[DistanceTrilinears[TrigExpand[LineEquation[AntiorthicAxis]/.Thread[{α,β,γ}->Trilinears[#]]]],TriangleQ[]]&/@ctrs//Timing

Radical Line

t=RandomTriangle[20];

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

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

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

Coaxal Systems

{Length[circs],circs}

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

Check Collinearity

Refine[Together//@ToSidelengths[CollinearQ[Center/@{ApolloniusCircle,ExcirclesRadicalCircle,NinePointCircle}]/.cos_Cos:>TrigExpand[cos]],TriangleQ[]]

Factor/@Refine[Together//@ToSidelengths[CollinearQ[Center/@{BevanCircle,Circumcircle,NinePointCircle}]/.cos_Cos:>TrigExpand[cos]],TriangleQ[]]

Check Radical Axis

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

Figure

t=Triangle[{{0,0},{1,0},{.2,.5}}];
Show[Graphics[{
t,
{Blue,ApolloniusCircle[t],NinePointCircle[t],ExcirclesRadicalCircle[t]},
{DarkGreen,BevanCircle[t],Circumcircle[t]},
{
Red,
l1=AntiorthicAxis[t],
{
DarkGreen,
l2=PerpendicularLine[Circumcenter[t],l1,10],
PerpSquare[{l1,l2},.15,3]
},
{
Blue,
l3=PerpendicularLine[X[970][t],l1,10],
PerpSquare[{l1,l3},.15,4]

}
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@{{X[970][t],"(Subscript[X, 970])"},{NinePointCenter[t],"F",LabelOffset->{0,.1}},
{Center[ExcirclesRadicalCircle[t]],"R"}},
Append[#,DarkGreen]&/@{{BevanPoint[t],"V"},{Circumcenter[t],"O",LabelOffset->{-.02,-.15}}}
],PointSize[.015],LabelOffset->.1]
}],AspectRatio->Automatic,PlotRange->{{-2.2,3.5},{-2.3,2.3}},
ImageSize->400]