Longuet-Higgins Circle

Author

Eric W. Weisstein
November 17, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/Longuet-HigginsCircle.nb.

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

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

Figure

circs={Circle[{1,0,0},b+c],Circle[{0,1,0},c+a],Circle[{0,0,1},a+b]};

t=Triangle[{{0,0},{1,0},{-.3,.6}}];
Show[Graphics[{
t,
{Blue,#[t]&/@circs},
{Red,LonguetHigginsCircle[t]},
PointLabels[Join[
Append[#,Blue]&/@Transpose[{VertexPoints[t],{"A","B","C"}}],
{{X[962][t],"(Subscript[X, 962])",Red}}
],LabelOffset->.2]
}],AspectRatio->Automatic]

Derivation

Symmetrize[Together//@(ToSidelengths[RadicalCenter[{Circle[{1,0,0},b+c],Circle[{0,1,0},c+a],Circle[{0,0,1},a+b]}]]/.Sqrt[s_]Sqrt[t_]:>Simplify[Sqrt[s t],TriangleQ[]])]

Through[{Name,Trilinears}[X[962]]]

Derivation of R.C.

rc=RadicalCircle[circs];

(ctr=Symmetrize[Together//@(ToSidelengths[Center[rc]]/.Sqrt[s_]Sqrt[t_]:>Simplify[Sqrt[s t],TriangleQ[]])])//Timing

Factor[ToSidelengths[CircleRadius[rc]^2]/.Sqrt[s_]Sqrt[t_]:>Simplify[Sqrt[s t],TriangleQ[]]]

radius=Refine[Sqrt[ToAngles[%64]],Area>0]

Refine[ToAngles[(1/(4 Area))(√((a^6)+2 (a^5) b-(a^4) (b^2)-4 (a^3) (b^3)-(a^2) (b^4)+2 a (b^5)+(b^6)+2 (a^5) c+2 (a^4) b c-4 (a^3) (b^2) c-4 (a^2) (b^3) c+2 a (b^4) c+2 (b^5) c-(a^4) (c^2)-4 (a^3) b (c^2)+10 (a^2) (b^2) (c^2)-4 a (b^3) (c^2)-(b^4) (c^2)-4 (a^3) (c^3)-4 (a^2) b (c^3)-4 a (b^2) (c^3)-4 (b^3) (c^3)-(a^2) (c^4)+2 a b (c^4)-(b^2) (c^4)+2 a (c^5)+2 b (c^5)+(c^6)))],Area>0]

Factor[ToSidelengths[CircleFunction[Circle[ctr,radius]]]]

KimberlingLookup[CyclicTrilinears[%79]]

Name[X[1500]]