Bevan Circle

Author

Eric W. Weisstein
November 21, 2004

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

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

Equation

Simplify[DistanceTrilinears[CircleEquation[X[40],2Circumradius[]]],TriangleQ[]]

Factor[%]

bevan=a (α^2)+a α β+b α β+c α β+b (β^2)+a α γ+b α γ+c α γ+a β γ+b β γ+c β γ+c (γ^2)

Simplify[bevan]

Passing Through Excenters

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

Simplify[Symmetrize[Numerator[%]]]

Graphic

{#,#/.X[n_]:>Subscript[X,n]}&/@{X[40],X[1054],X[1282],X[1768]}

t=Triangle[{{0,0},{1,0},{.3,.5}}];
Show[Graphics[
{
t,
{Red,BevanCircle[t]},
{Green,ExcentralTriangle[t]},
{Blue,Excircles[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Violet]&/@({#[t],#/.X[n_]:>Subscript["X",n]}&/@{X[1054],X[1282],X[1768]}),
Append[#,Blue]&/@Transpose[{Excenters[t],{"(Subscript[J, A])","(Subscript[J, B])","(Subscript[J, C])"}}],
{{X[40][t],"V",Red}}
],LabelOffset->.07
]
}],AspectRatio->Automatic]

Circle Function

CircleFunction[BevanCircle]

Center

CenterFunction[X[40]]

CenterName[X[40]]

CircleCenter[BevanCircle]

Circumradius

Simplify[Together//@DistanceTrilinears[CircleRadius[BevanCircle]],TriangleQ[]]

R=CircleRadius[TrilinearVertexMatrix[ExcentralTriangle]];//Timing

Factor[Together//@R]//Timing

Incident Centers

IncidentCenters[KimberlingCenters,BevanCircle]//Timing

CenterName/@{X[1054],X[1282],X[1768]}

Orthogonality

Check

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

Graphic

t=Triangle[{{0,0},{1,0},{.1,.7}}];
circs={BevanCircle,StevanovicCircle};
centers={"(Subscript[X, 40])","(Subscript[X, 650])"};
colors={Red,Green};
Show[Graphics[{
Transpose[{colors,Through[circs[t]]}],
PointLabels[Join[
Transpose[{CircleCenter[#[t]]&/@circs,centers,colors}]
],LabelOffset->.2
]
}],AspectRatio->Automatic]