Moses–Longuet-Higgins Circle

Author

Eric W. Weisstein
November 17, 2005

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

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

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

Figure

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

t=Triangle[{{0,0},{1,0},{-.1,.6}}];
Show[Graphics[{
t,
{Blue,#[t]&/@circs},
{Red,circ=MosesLonguetHigginsCircle[t]},
PointLabels[Join[
Append[#,Blue]&/@Transpose[{VertexPoints[t],{"A","B","C"}}],
{{Center[circ],"(Subscript[O, M - L - H])",Red}}
],LabelOffset->.2]
}],AspectRatio->Automatic]

Derivation

Symmetrize[Together//@(ToSidelengths[RadicalCenter[circs]]/.Sqrt[s_]Sqrt[t_]:>Simplify[Sqrt[s t],TriangleQ[]])]

KimberlingLookup[%]

Symmetrize[ToSidelengths[(a^2 (SA-4 sa^2)+4 (sc^2 SB+sb^2 SC))/a]]

Factor[CircleRadius[RadicalCircle[circs]]^2]

Factor[ToSidelengths[9 Circumradius^2 + 4 Inradius^2 (1 - (12 (a + b + c) Circumradius^2) / (a b c))]]

Factor[ToSidelengths[CircleFunction[RadicalCircle[circs]]]]

KimberlingLookup[%]

Name[X[220]]