Soddy Line

Author

Eric W. Weisstein
March 20, 2006

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

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

©2006 Wolfram Research, Inc. except for portions noted otherwise

Graphic

pts=Partition[{
OuterGriffithsPoint,"Gr'",
EppsteinPoint[1],"(Subscript[Ep, 1])",
OuterRigbyPoint,"Ri'",
OuterSoddyCenter,"S'",
Incenter,"I",
InnerSoddyCenter,"S",
InnerRigbyPoint,"Ri",
EppsteinPoint[2],"Ol",
InnerGriffithsPoint,"Gr",
GergonnePoint,"Ge",
DeLongchampsPoint,"Z",
FletcherPoint,"Fl"
},2];

t=Triangle[{{0,0},{1,0},{.35,.7}}];
Show[Graphics[{
t,
{Red,LongestLine[Coordinates/@(First[#][t]&/@pts),2]},
PointLabels[
{#1[t],#2}&@@@pts,
LabelOffset->.04
]
}],AspectRatio->Automatic]

Equation

LineEquation[SoddyLine]

LineFunction[SoddyLine]

Center

Off[Divide::"infy",∞::"indet",Power::"infy"]
IncidentCenters[KimberlingCenters,CyclicTrilinears[LineFunction[SoddyLine]]]//Timing

Incident Centers

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

First/@ctrs

Length[ctrs]

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

Check

Do[
Print[{i,ctrs[[i]],
TimeConstrained[FullSimplify[Simplify[DistanceTrilinears[TrigExpand[LineEquation[SoddyLine]/.Thread[{α,β,γ}->Trilinears[ctrs[[i]]]]]],TriangleQ[]],TriangleQ[]],100]//Timing}],
{i,Length[ctrs]}]

Simplify[DistanceTrilinears[LineEquation[SoddyLine]/.Thread[{α,β,γ}->Trilinears[#]]/.{Sec[x_/2]:>Sqrt[2/(1+Cos[x])],Tan[x_/2]:>Sin[x]/(1+Cos[x])}],TriangleQ[]]&/@{X[170],X[279]}//Timing

Perpendicular to the Gergonne Line

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

Radical Line

t=RandomTriangle[20];

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

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

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

Coaxal Systems

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

{Length[circs],circs}

circs={EulerGergonneSoddyCircle,GEOSCircle};

Check Collinearity

CollinearQ[Center/@circs]

Check Radical Axis

Simplify[Together//@ToSidelengths[First/@(RadicalLine@@@coaxal)],TriangleQ[]]

Figure

t=Triangle[{{0,0},{1,0},{.75,.4}}];
Show[Graphics[{
t,
{Blue,Through[circs[t]]},
{
Red,
l1=SoddyLine[t],
{
Dashing[{.015}],
l2=PerpendicularLine[Center[GEOSCircle[t]],l1,10]
},
PerpSquare[{l1,l2},.15,4]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@{
{Center[GEOSCircle[t]],"(Subscript[X, GEOS])"},
{Center[EulerGergonneSoddyCircle[t]],"(Subscript[X, EGS])"}
}
],PointSize[.02],LabelOffset->.1]
}],AspectRatio->Automatic,PlotRange->{{-.5,2.},{-1,1}}]