Johnson-Yff Circles

Author

Eric W. Weisstein
January 20, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/Johnson-YffCircles.nb.

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[GraphicsArray[{
Graphics[{
t,
{Blue,YffCircles[1][t]},
{Red,YffJohnsonCircle[1][t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{Center/@YffCircles[1][t],{"(Subscript[Y, A])","(Subscript[Y, B])","(Subscript[Y, C])"}}],
{
{Center[YffJohnsonCircle[1][t]],"(Subscript[X, 1478])",Red}
}
],LabelOffset->.03]
},AspectRatio->Automatic],
Graphics[{
t,
{Blue,YffCircles[2][t]},
{Red,YffJohnsonCircle[2][t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{Center/@YffCircles[2][t],{"(Subscript[Z, A])","(Subscript[Z, B])","(Subscript[Z, C])"}}],
{
{Center[YffJohnsonCircle[2][t]],"(Subscript[X, 1479])",Red},
}
],LabelOffset->.06,PointSize[.025]]
},AspectRatio->Automatic]
}],ImageSize->500,GraphicsSpacing->-.3]

Centers

CenterName/@{X[1478],X[1479]}

CenterFunction/@{X[1478],X[1479]}

Radius

CircleRadius/@JohnsonYffCircle/@{1,2}//TraditionalForm

Circle Functions

CircleFunction[JohnsonYffCircle[1]]//TraditionalForm

IncidentCenters[KimberlingCenters,CyclicTrilinears[CircleFunction[JohnsonYffCircle[1]]]]//Timing

CircleFunction[JohnsonYffCircle[2]]//TraditionalForm

IncidentCenters[KimberlingCenters,CyclicTrilinears[CircleFunction[JohnsonYffCircle[2]]]]//Timing

Incident Centers

First Circle

(ctrs=IncidentCenters[KimberlingCenters,JohnsonYffCircle[1]])//Timing

First/@ctrs

Length[ctrs]

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

Second Circle

(ctrs=IncidentCenters[KimberlingCenters,JohnsonYffCircle[2]])//Timing

First/@ctrs

Length[ctrs]

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

Derivation

1+2Cos[B]Cos[C]

Factor[ToSidelengths[1 + 2 Cos[ B] Cos [C ]]]

ToAngles[-%369]

-(a^4 - b^4 + 2*a^2*b*c + 2*b^2*c^2 - c^4)//InputForm

{a^4-b^4+2*a^2*b*c+2*b^2*c^2-c^4,2a^2b c(1+2Cos[B]Cos[C])}

eqn=Factor[Factor[Together//@ToSidelengths[TrilinearEquation[Center[JohnsonYffCircle[1]],CircleRadius[JohnsonYffCircle[1]]]]]/.Sqrt[x_]Sqrt[y_]:>FullSimplify[Sqrt[x y],TriangleQ[]]]

Numerator[Symmetrize[eqn]]

l=Factor[CircleFunction[Numerator[Symmetrize[eqn]]][[1]]]

InputForm[l]

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

eqn=Factor[Factor[Together//@ToSidelengths[TrilinearEquation[Center[JohnsonYffCircle[2]],CircleRadius[JohnsonYffCircle[2]]]]]/.Sqrt[x_]Sqrt[y_]:>FullSimplify[Sqrt[x y],TriangleQ[]]]

Numerator[Symmetrize[eqn]]

l=Factor[CircleFunction[Numerator[Symmetrize[eqn]]][[1]]]

InputForm[l]

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