Extangents Circle

Author

Eric W. Weisstein
January 15, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.35,.7}}];
Show[Graphics[{
{Red,ExtangentsCircle[t]},
{Green,Excircles[t]},
{Blue,ExtangentsTriangle[t]},
t,
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Green]&/@Transpose[{Center/@Excircles[t],{"(Subscript[J, A])","(Subscript[J, B])","(Subscript[J, C])"}}],
Append[#,Red]&/@Transpose[{VertexPoints[ExtangentsTriangle[t]],{"A'","B'","C'"}}],
{
{Center[ExtangentsCircle[t]],"X",Red}
}
],LabelOffset->.12]
}],AspectRatio->Automatic]

Radius

CircleRadius[ExtangentsCircle]//TraditionalForm

Center

Center[ExtangentsCircle][[1]]//TraditionalForm

Circle Function

CircleFunction[ExtangentsCircle]//TraditionalForm

Relationship to Euler Line and Intangents Circle

Factor[#/PolynomialGCD@@#]&[Factor[Midpoint[Center/@{ExtangentsCircle,IntangentsCircle}]]]

IncidentCenters[{X[26]},%170]

Symmetrize[Together//@ToSidelengths[CenterFunction[X[26]]/.Cos[x_]:>TrigExpand[Cos[x]]]]

CenterName[X[26]]

t=Triangle[{{0,0},{1,0},{.4,.7}}];
lines=PerpendicularLine[#[t],EulerLine[t]]&/@{X[1],X[40]};
Show[Graphics[{
t,
{Red,EulerLine[t]},
{Green,Line[Coordinates/@{X[26][t],X[55][t]}]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{
{Center[IntangentsCircle[t]],"(Subscript[O, I])",Blue},
{Center[ExtangentsCircle[t]],"(Subscript[O, E])",Blue}
}
],
LabelOffset->.05
]
}],AspectRatio->Automatic,PlotRange->{{-.1,2},{-1,1}}]

Incident Points

IncidentCenters[KimberlingCenters,ExtangentsCircle]//Timing