Lucas Inner Circle

Author

Eric W. Weisstein
October 21, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{Red,LucasCircles[t]},
{Green,LucasCirclesRadicalCircle[t]},
{Blue,Circumcircle[t],LucasInnerCircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"B","C","A"}}],
Append[#,Red]&/@Transpose[{Center/@LucasCircles[t],{"(Subscript[L, B])","(Subscript[L, C])","(Subscript[L, A])"}}]
],LabelOffset->.03,PointSize[.025]]
}],AspectRatio->Automatic,PlotRange->All]

Center

Center[LucasInnerCircle][[1]]//TraditionalForm

Radius

CircleRadius[LucasInnerCircle]//TraditionalForm

Circle Function

Factor[Together//@ToSidelengths[TrilinearEquation[Circle[LucasInnerCircle]]]/.Sqrt[x_]Sqrt[y_]:>Simplify[Sqrt[x y],TriangleQ[]]/.Sqrt[x_]:>Simplify[Sqrt[x],TriangleQ[]]]

eqn=(-4192 (a^7) b c (α^2)+6240 (a^5) (b^3) c (α^2)+6240 (a^3) (b^5) c (α^2)-4192 a (b^7) c (α^2)+6240 (a^5) b (c^3) (α^2)+31296 (a^3) (b^3) (c^3) (α^2)+6240 a (b^5) (c^3) (α^2)+6240 (a^3) b (c^5) (α^2)+6240 a (b^3) (c^5) (α^2)-4192 a b (c^7) (α^2)-56 (a^5) b c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (α^2)+10864 (a^3) (b^3) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (α^2)-56 a (b^5) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (α^2)+10864 (a^3) b (c^3) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (α^2)+10864 a (b^3) (c^3) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (α^2)-56 a b (c^5) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (α^2)-2145 (a^8) c α β+10628 (a^6) (b^2) c α β-12870 (a^4) (b^4) c α β+10628 (a^2) (b^6) c α β-2145 (b^8) c α β+14820 (a^6) (c^3) α β-12772 (a^4) (b^2) (c^3) α β-12772 (a^2) (b^4) (c^3) α β+14820 (b^6) (c^3) α β-19110 (a^4) (c^5) α β-37828 (a^2) (b^2) (c^5) α β-19110 (b^4) (c^5) α β+4388 (a^2) (c^7) α β+4388 (b^2) (c^7) α β+2047 (c^9) α β+3640 (a^6) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α β-56 (a^4) (b^2) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α β-56 (a^2) (b^4) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α β+3640 (b^6) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α β-21952 (a^2) (b^2) (c^3) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α β-10920 (a^2) (c^5) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α β-10920 (b^2) (c^5) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α β+3696 (c^7) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α β-4192 (a^7) b c (β^2)+6240 (a^5) (b^3) c (β^2)+6240 (a^3) (b^5) c (β^2)-4192 a (b^7) c (β^2)+6240 (a^5) b (c^3) (β^2)+31296 (a^3) (b^3) (c^3) (β^2)+6240 a (b^5) (c^3) (β^2)+6240 (a^3) b (c^5) (β^2)+6240 a (b^3) (c^5) (β^2)-4192 a b (c^7) (β^2)-56 (a^5) b c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (β^2)+10864 (a^3) (b^3) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (β^2)-56 a (b^5) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (β^2)+10864 (a^3) b (c^3) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (β^2)+10864 a (b^3) (c^3) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (β^2)-56 a b (c^5) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (β^2)-2145 (a^8) b α γ+14820 (a^6) (b^3) α γ-19110 (a^4) (b^5) α γ+4388 (a^2) (b^7) α γ+2047 (b^9) α γ+10628 (a^6) b (c^2) α γ-12772 (a^4) (b^3) (c^2) α γ-37828 (a^2) (b^5) (c^2) α γ+4388 (b^7) (c^2) α γ-12870 (a^4) b (c^4) α γ-12772 (a^2) (b^3) (c^4) α γ-19110 (b^5) (c^4) α γ+10628 (a^2) b (c^6) α γ+14820 (b^3) (c^6) α γ-2145 b (c^8) α γ+3640 (a^6) b (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α γ-10920 (a^2) (b^5) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α γ+3696 (b^7) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α γ-56 (a^4) b (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α γ-21952 (a^2) (b^3) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α γ-10920 (b^5) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α γ-56 (a^2) b (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α γ+3640 b (c^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) α γ+2047 (a^9) β γ+4388 (a^7) (b^2) β γ-19110 (a^5) (b^4) β γ+14820 (a^3) (b^6) β γ-2145 a (b^8) β γ+4388 (a^7) (c^2) β γ-37828 (a^5) (b^2) (c^2) β γ-12772 (a^3) (b^4) (c^2) β γ+10628 a (b^6) (c^2) β γ-19110 (a^5) (c^4) β γ-12772 (a^3) (b^2) (c^4) β γ-12870 a (b^4) (c^4) β γ+14820 (a^3) (c^6) β γ+10628 a (b^2) (c^6) β γ-2145 a (c^8) β γ+3696 (a^7) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) β γ-10920 (a^5) (b^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) β γ+3640 a (b^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) β γ-10920 (a^5) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) β γ-21952 (a^3) (b^2) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) β γ-56 a (b^4) (c^2) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) β γ-56 a (b^2) (c^4) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) β γ+3640 a (c^6) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) β γ-4192 (a^7) b c (γ^2)+6240 (a^5) (b^3) c (γ^2)+6240 (a^3) (b^5) c (γ^2)-4192 a (b^7) c (γ^2)+6240 (a^5) b (c^3) (γ^2)+31296 (a^3) (b^3) (c^3) (γ^2)+6240 a (b^5) (c^3) (γ^2)+6240 (a^3) b (c^5) (γ^2)+6240 a (b^3) (c^5) (γ^2)-4192 a b (c^7) (γ^2)-56 (a^5) b c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (γ^2)+10864 (a^3) (b^3) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (γ^2)-56 a (b^5) c (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (γ^2)+10864 (a^3) b (c^3) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (γ^2)+10864 a (b^3) (c^3) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (γ^2)-56 a b (c^5) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]) (γ^2));

FullSimplify[Factor[CircleFunction[eqn][[1]]],TriangleQ[]]

TraditionalForm[Factor[%41/.(Sqrt[-((a-b-c) (a+b-c) (a-b+c) (a+b+c))]):>Area]]

Orthogonal Circles

t=Triangle[{{0,0},{1,0},{.1,.7}}];
circs={LucasInnerCircle,ParryCircle};
centers={"(Subscript[X, LIS])","(Subscript[X, 351])"};
colors={Red,Green};
Show[Graphics[{
t,
Transpose[{colors,Through[circs[t]]}],
PointLabels[Join[
Transpose[{Center[#[t]]&/@circs,centers,colors}],
Transpose[{VertexPoints[t],{"A","B","C"}}]
],LabelOffset->.07
]
}],AspectRatio->Automatic,PlotRange->{{-.7,1.1},{-.7,1}}]