Griffiths Points

Author

Eric W. Weisstein
May 19, 2003

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

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

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

Initializtaion

<<PlaneGeometry`;
<<KimberlingCenters`;

Figure

Figure

t=Triangle[{{0,0},{1,0},{.4,.7}}];
s1=InnerSoddyTriangle[t];
s2=OuterSoddyTriangle[t];

t1=TangentialTriangle[s1];
t2=TangentialTriangle[s2];

Show[GraphicsArray[{
Graphics[{
t,
{Red,t2},
{Blue,s1},
{Orange,Line/@Transpose[Vertices/@{s1,t2}]},
{Green,TangentCircles[t]},
{Green,SoddyCircles[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[s1],{"P","Q","R"}}],
Append[#,Red]&/@Transpose[{VertexPoints[t2],{"X'","Y'","Z'"}}],
{
{InnerGriffithsPoint[t],"Gr",Orange}
}
],LabelOffset->.03,PointSize[.025]]
},
AspectRatio->Automatic],
Graphics[{
t,
{Red,t1},
{Blue,s2},
{Orange,Line/@Transpose[Vertices/@{s2,t1}]},
{Green,TangentCircles[t]},
{Green,SoddyCircles[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[s2],{"P'","Q'","R'"}}],
Append[#,Red]&/@Transpose[{VertexPoints[t1],{"X","Y","Z"}}],
{
{OuterGriffithsPoint[t],"Gr'",Orange}
}
],LabelOffset->.07,PointSize[.025]]
},
AspectRatio->Automatic]
}]]

Trilinears

FullSimplify[{1,1,1}+4Area[]/{a(s-a),b(s-b),c(s-c)}/.s->Semiperimeter[]/.(Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])->4Δ]

CenterFunction/@{InnerGriffithsPoint,OuterGriffithsPoint}

BisymmetricQ/@{InnerGriffithsPoint,OuterGriffithsPoint}

IncidentCenters[KimberlingCenters,#]&/@{InnerGriffithsPoint,OuterGriffithsPoint}