Euler-Gergonne-Soddy Triangle

Author

Eric W. Weisstein
November 8, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/Euler-Gergonne-SoddyTriangle.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Euler-Gergonne-SoddyTriangle.html.

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.5}}];
Show[Graphics[{
t,
{Red,s=SoddyLine[t]},
{Blue,EulerLine[t]},
{DarkGreen,g=GergonneLine[t]},
PerpSquare[{s,g},.08,1],
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,DarkGreen]&/@Transpose[{NobbsPoints[t],{"E'","F'","G'"}}],
{
{DeLongchampsPoint[t],"Z",Purple},
{FletcherPoint[t],"Fl",Orange},
{EvansPoint[t],"Ev",CornflowerBlue}
}
]]
}],AspectRatio->Automatic,PlotRange->{{-1.5,1.5},{-.5,1.5}}]

Gergonne Line and Soddy Line Are Perpendicular

ColumnForm[{{l,m,n},{ll,mm,nn}}=CyclicTrilinears[LineFunction[#]]&/@{SoddyLine,GergonneLine}]

DistanceTrilinears[(l ll+m mm+n nn)-(m nn+mm n)Cos[A]-(n ll+nn l)Cos[B]-(l mm+ll m)Cos[C]]//Expand

Trilinear Vertex Matrix

TrilinearVertexMatrix[EulerGergonneSoddyTriangle]//TraditionalForm

Area

Factor[Together//@ToSidelengths[HalfAngleFormulas[Area[EulerGergonneSoddyTriangle]]]]

Edge Lengths

Factor[Together//@ToSidelengths[HalfAngleFormulas[SideLengths[EulerGergonneSoddyTriangle]]]]

Centers

(pts={#,KimberlingLookup[#[EulerGergonneSoddyTriangle]]}&/@Take[KimberlingCenters,100]);//Timing

{#[[1]],Name[#[[1]]],X[#[[2,1]]],Name[X[#[[2,1]]]]}&/@Cases[pts,_?(Last[#]=!={}&)]//TableForm