Gergonne Line

Author

Eric W. Weisstein
October 21, 2005

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

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

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

Figure

tri={{0,0},{1,-0.1},{0.4,0.39}};
pts={
	Orthocenter,"H",Circumcenter,"O",EvansPoint,"Ev",
	Centroid,"G",FletcherPoint,"Fl",SoddyCenterInner,"S",
	SoddyCenterOuter,"S'",GergonnePoint,"Ge",
	nobbs[1],"F'",nobbs[2],"D'",nobbs[3],"E'"
};

Show[Graphics[{
			Triangle[tri] ,
			PointLabels[tri,pts],
			Line[LongestLine[{SoddyCenterInner[tri],SoddyCenterOuter[tri],
					FletcherPoint[tri]} /. Point->Identity]],
			Line[LongestLine[{Incenter[tri],Orthocenter[tri],Circumcenter[tri],
					EvansPoint[tri]} /. Point->Identity]],
			Line[LongestLine[NobbsPoints[tri]]] 
			}],
	AspectRatio->Automatic,PlotRange->All]

Equation

LineEquation[GergonneLine]

Perpendicular to the Soddy Line

Together//@ToSidelengths[PerpendicularQ@@(TrilinearLine/@{GergonneLine,SoddyLine})]

Incident Centers

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

First/@ctrs

Length[ctrs]

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

{#,CenterName[#]}&/@{X[101]}//TableForm

Check

Do[
Print[{i,ctrs[[i]],
FullSimplify[Simplify[DistanceTrilinears[TrigExpand[LineEquation[GergonneLine]/.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[GergonneLine][[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

Together//@ToSidelengths[CollinearQ[Center/@(circs1={BevanCircle,ExcirclesRadicalCircle,PolarCircle})]]

CollinearQ[Center/@(circs2={Incircle,InnerSoddyCircle,OuterSoddyCircle})]

Check Radical Axis

Factor[Together//@ToSidelengths[First/@ExactTrilinears/@ToSidelengths[First/@(RadicalLine@@@coaxal)]]]

Figure

t=Triangle[{{0,0},{1,0},{.3,.5}}];
Show[Graphics[{
t,
{Red,l1=GergonneLine[t]},
{
Blue,
Through[circs1[t]],
l2=PerpendicularLine[Center[circs1[[1]][t]],l1,10],
PerpSquare[{l1,l2},.15,4]
},
{
DarkGreen,
Through[circs2[t]],
l3=PerpendicularLine[Center[circs2[[1]][t]],l1,10],
PerpSquare[{l1,l3},.15,3]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@{
{InnerSoddyCenter[t],"S"},
{OuterSoddyCenter[t],"S'"}
}
],PointSize[.015],LabelOffset->.1]
}],AspectRatio->Automatic,PlotRange->{{-2.4,3.5},{-2.3,2.7}}]