Tangential Circle

Author

Eric W. Weisstein
November 29, 2004

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

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

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

Derivation

Factor[Together//@CircleEquation[TrilinearVertexMatrix[TangentialTriangle]]]

Factor[Cancel[CircleFunction[Symmetrize[Numerator[%31]]]]]

InputForm[%]

Circle Function

CircleFunction[TangentialCircle]

Symmetrize[LawOfCosines[Symmetrize[%]]]

IncidentCenters[KimberlingCenters,CyclicTrilinears[Cos[A]]]//Timing

Through[{CenterName,CenterFunction}[X[3]]]

Circle Radius

FullSimplify[Together//@DistanceTrilinears[CircleRadius[TangentialCircle]],TriangleQ[]]

Cancel[Simplify[LawOfCosines[%23],TriangleQ[]](R/(Circumradius[]))]

Circle Center

Symmetrize[Factor[Together//@DistanceTrilinears[CircleCenter[TangentialCircle][[1]]]]]

CenterName[X[26]]

CenterFunction[X[26]]

Symmetrize[Together//@DistanceTrilinears[TrigExpand[CenterFunction[X[26]]]]]

Symmetrize[LawOfCosines[%]]

IncidentCenters[KimberlingCenters,CircleCenter[TangentialCircle]]//Timing

Incident Centers

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

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

Check

Short/@(FullSimplify[DistanceTrilinears[TrigExpand[eqn/.Thread[{α,β,γ}->Trilinears[#]]]],
TriangleQ[]]&/@ctrs)//Timing

Orthogonality

Check

Together//@DistanceTrilinears[First[OrthogonalQ[TangentialCircle,StevanovicCircle]]]//Timing

Graphic

t=Triangle[{{0,0},{1,0},{.1,.7}}];
circs={TangentialCircle,StevanovicCircle};
centers={"(Subscript[X, 26])","(Subscript[X, 650])"};
colors={Red,Green};
Show[Graphics[{
Transpose[{colors,Through[circs[t]]}],
PointLabels[Join[
Transpose[{CircleCenter[#[t]]&/@circs,centers,colors}]
],LabelOffset->.5
]
}],AspectRatio->Automatic,PlotRange->All]

Figure

t=Triangle[{{0,0},{1,0},{.1,.5}}];
Show[Graphics[{
t,
{Red,TangentialTriangle[t]},
{Blue,TangentialCircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[TangentialTriangle[t]],{"(Subscript[T, A])","(Subscript[T, B])","(Subscript[T, B])"},{{},{LabelOffset->{-.25,0}},{}}}],
{{X[26][t],"(Subscript[X, 26])",Blue}}
],{PointSize[.025],LabelOffset->.17}]
}],AspectRatio->Automatic,ImageSize->200,PlotRange->All]