Incentral Circle

Author

Eric W. Weisstein
January 16, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{Red,IncentralCircle[t]},
{Blue,IncentralTriangle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[IncentralTriangle[t]],{"(Subscript[I, A])","(Subscript[I, B])","(Subscript[I, C])"}}],
{
{Center[IncentralCircle[t]],"(Subscript[O, I])",Red}
}
],LabelOffset->.04,PointSize[.03]]
}],AspectRatio->Automatic,PlotRange->All]

Radius

CircleRadius[IncentralCircle]//TraditionalForm

CircleRadius[IncentralCircle]

TraditionalForm[area=Refine[(√(-(a b c ((a^3)-(a^2) b-a (b^2)+(b^3)+(a^2) c-3 a b c+(b^2) c-a (c^2)-b (c^2)-(c^3)) ((a^3)+(a^2) b-a (b^2)-(b^3)+(a^2) c+3 a b c+(b^2) c-a (c^2)+b (c^2)-(c^3)) ((a^3)+(a^2) b-a (b^2)-(b^3)-(a^2) c-3 a b c-(b^2) c-a (c^2)+b (c^2)+(c^3)))/(-(4Area)^2)))/(2 (a+b) (a+c) (b+c)),Area>0]]

IncentralCircle[t]

ToSidelengths[area]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

Center

ToAngles[Factor[Center[IncentralCircle][[1]]]]

IncidentCenters[KimberlingCenters,Center[IncentralCircle]]//Timing

Circle Function

l=ToAngles[Factor[CircleFunction[IncentralCircle]]]

IncidentCenters[KimberlingCenters,CyclicTrilinears[l]]//Timing

CenterName/@Last[%28]

CenterFunction[X[191]]

Simplify[Plus@@{(a^3)+(a^2) b-a (b^2)-(b^3)+(a^2) c-a b c-(b^2) c-a (c^2)-b (c^2)-(c^3),-(a^3)+(b^3)+(c^3)+(-a+b+c) (a b+a c+b c)}]

Incident Centers

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

First/@ctrs

Length[ctrs]

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

Derivation

circ1=Together//@Circumcircle@@TrilinearVertexMatrix[IncentralTriangle];

circ=circ1/.Circle[ctr_List,r_]:>Circle[Symmetrize/@ctr,FullSimplify[r,TriangleQ[]]]

Factor//@Refine[CircleRadius[circ],TriangleQ[]]

InputForm[%]

Center[circ][[1]]//InputForm

IncidentCenters[KimberlingCenters,Center[circ]]//Timing

eqn=Symmetrize[Numerator[Together//@ToSidelengths[TrilinearEquation[circ]]/.Sqrt[x_]Sqrt[y_]:>Simplify[Sqrt[x y],TriangleQ[]]]]

CircleFunction[eqn]

InputForm[CircleFunction[eqn][[1]]]

IncidentCenters[KimberlingCenters,%]//Timing

CenterName[X[191]]

t=Triangle[{{0,0},{1,0},{.3,.5}}];
Show[Graphics[{
t,
{Blue,IncentralTriangle[t]},
{Red,IncentralCircle[t]}
}],AspectRatio->Automatic]