Power Circle

Author

Eric W. Weisstein
December 8, 2004

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

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

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

Derivation

m=Midpoint/@Partition[TrilinearVertexMatrix[ReferenceTriangle],2,1,1]

v=RotateRight[TrilinearVertexMatrix[ReferenceTriangle]]

rs=Simplify[Distance@@@Transpose[{m,v}],TriangleQ[]]

CyclicTrilinears[(1/2) (Sqrt[2 (a^2)+2 (b^2)-(c^2)])]

CircleTriplet=.

CircleTriplet[ctr_,r_][t_Triangle]:=Module[{ctrs=MapIndexed[RotateRight[#1,#2[[1]]-1]&,CyclicTrilinears[ctr]]},
Circle@@@Transpose[{TrilinearToCartesian[t,#]&/@ctrs,CyclicTrilinears[r]}]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]/.Thread[{A,B,C}->Angles[t]]
]

PowerCircles[t_Triangle]:=Circle@@@Transpose[{TrilinearToCartesian[t,#]&/@{{b,a,0},{0,c,b},{c,0,a}},{(1/2) (Sqrt[2 (a^2)+2 (b^2)-(c^2)]),(1/2) (Sqrt[-(a^2)+2 ((b^2)+(c^2))]),(1/2) (Sqrt[2 (a^2)-(b^2)+2 (c^2)])}}]/.sides;

Simplify[Together//@ToSidelengths[CircleEquation[m[[1]],rs[[1]]]],TriangleQ[]]

Factor[%]

eqn=((a^4) (α^2)+(a^2) (b^2) (α^2)-(a^2) (c^2) (α^2)+2 (a^3) b α β+2 a (b^3) α β+(a^2) (b^2) (β^2)+(b^4) (β^2)-(b^2) (c^2) (β^2)+(a^3) c α γ+3 a (b^2) c α γ-a (c^3) α γ+3 (a^2) b c β γ+(b^3) c β γ-b (c^3) β γ);

Factor/@Coefficient[eqn,#]&/@{(α^2),(β^2),(γ^2),α β,α γ,β γ}

Symmetrize/@(Together//@ToSidelengths[CircleCenter[eqn]])

ToAngles[Factor[CircleFunction[eqn]]]

{{b,a,0},{0,c,b},{c,0,a}}

Graphic

t=Triangle[{{0,0},{1,0},{.3,.5}}];
colors={Red,Blue,Green};
Show[Graphics[{
t,
Transpose[{colors,Through[PowerCircles[t]]}],
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"},colors}],
Transpose[{VertexPoints[MedialTriangle[t]],{"(Subscript[M, A])","(Subscript[M, B])","(Subscript[M, C])"},colors}]
]]

}],AspectRatio->Automatic]

Incident Centers

Timing[IncidentCenters[KimberlingCenters,CircleEquation[#]]]&/@Circles[PowerCircles]

Table[IncidentCenters[{X[134]},CircleEquation[Circles[PowerCircles][[3]]]],{10}]

Radical Circle

RadicalCircle[c:{_Circle,_Circle,_Circle}]:=Module[{l=CircleFunction/@CircleEquation/@c,r,ctr},
ctr=RadicalCenter[l,{1,1,1}];
Circle[ctr,Sqrt[Distance[ctr,CircleCenter[l[[1]]]]^2-CircleRadius[l[[1]]]^2]]
]

RadicalCircle[PowerCircles]

eqns=Symmetrize/@Simplify[Together//@ToSidelengths[CircleEquation/@PowerCircles],TriangleQ[]]

CircleFunction/@eqns

ctr=Symmetrize/@RadicalCenter[%70,{1,1,1}]

Symmetrize/@(Together//@ToSidelengths[CircleCenter[DeLongchampsCircle]])

Factor[Together//@(Distance[ctr,CircleCenter[PowerCircles[[1]]]]^2-CircleRadius[PowerCircles[[1]]]^2)]

Factor[ToSidelengths[CircleRadius[DeLongchampsCircle]^2]]