Intangents Circle

Author

Eric W. Weisstein
December 19, 2004

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.35,.7}}];
Show[Graphics[{
t,
{Green,Incircle[t]},
{Red,IntangentsCircle[t]},
{Blue,IntangentsTriangle[t],
{Dashing[{.02}],Sides[IntangentsTriangle[t],1]}},
{Purple,Excircles[t],{Dashing[{.02}],Sides[t,5]}},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Purple]&/@Transpose[{VertexPoints[ExcentralTriangle[t]],{"(Subscript[J, A])","(Subscript[J, B])","(Subscript[J, C])"}}],
Append[#,Red]&/@Transpose[{VertexPoints[IntangentsTriangle[t]],{"A'","B'","C'"}}],
{
{Incenter[t],"I",Green},
{Center[IntangentsCircle[t]],"X",Red}
}
],LabelOffset->.05]
}],AspectRatio->Automatic,PlotRange->{{-.5,1.3},{-.4,1.5}}]

Radius

CircleRadius[IntangentsCircle]//TraditionalForm

Center Function

Center[IntangentsCircle]//TraditionalForm

Circle Function

Factor[ToAngles[CircleFunction[IntangentsCircle]]]

Incident Points

IncidentCenters[KimberlingCenters,IntangentsCircle]//Timing

Derivation

v=Map[Symmetrize,Together//@ToSidelengths[TrilinearVertexMatrix[IntangentsTriangle]],{2}]

eqn=(Together//@(TrilinearEquation@@v/.{s->(Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]),(a-b-c) (a+b-c) (a-b+c) (a+b+c)->(-s^2)}))

eqn2=Factor[%3/.s^n_:>((a+b-c) (a-b+c) (-a+b+c) (a+b+c))^(n/2)];

eqn=eqn2[[14]]

lmn=Factor[CircleFunction[eqn]]

IncidentCenters[KimberlingCenters,lmn]//Timing

lmn[[1]]//InputForm

ctr=Symmetrize/@Center[eqn]

IncidentCenters[KimberlingCenters,ctr]//Timing

InputForm[ctr[[1]]]

r=Together//@ExpandAll[Together//@CircleRadius[eqn]]

FullSimplify[r,TriangleQ[]]

Refine[ToAngles[%],TriangleQ[]]

Inradius[]

Simplify[((Sqrt[-(((a-b-c) (a+b-c) (a-b+c))/(a+b+c))])/(8 Abs[Cos[A]] Abs[Cos[B]] Abs[Cos[C]]))(r/(Inradius[]))]

InputForm[%]

t=Triangle[{{0,0},{1,0},{.2,.7}}];
Show[Graphics[{
t,
{Blue,IntangentsTriangle[t]},
{Red,IntangentsCircle[t]}
}],AspectRatio->Automatic]