Lester Circle

Author

Eric W. Weisstein
October 21, 2005

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

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

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

Figure

tri=Triangle[{{0.,0.},{1.,.9},{2.,-1.6}}];
circum=Circumcenter[tri];
nine=NinePointCenter[tri];
f1=FermatPoint[1][tri];
f2=FermatPoint[2][tri];
pts={circum,C,nine,N,f1,(Subscript[F, 1]),f2,(Subscript[F, 2])};

Show[Graphics[{
tri,
Circumcircle[tri],
NinePointCircle[tri],
{Dashing[{.01,.02}],
Line/@EquilateralTriangularize[1,tri],Line/@EquilateralTriangularize[2,tri]},
{RGBColor[1,0,0],FermatPointConstruction[#,tri]&/@{1,2}},
{Thickness[.01],Circumcircle[{circum[[1]],nine[[1]],f1[[1]]}]},
PointLabels[tri,pts,.03]
}],AspectRatio->Automatic]

Circle Function

IncidentCenters[KimberlingCenters,CyclicTrilinears[CircleFunction[LesterCircle]]]//Timing

Circle Equation

CircleFunction[LesterCircle]

InputForm[%]

Derivation

Factor[Simplify[Together//@DistanceTrilinears[CircleEquation[Trilinears/@{Circumcenter,FermatPoint[1],FermatPoint[2]}]/.Csc[x_]:>TrigExpand[Csc[x]]],TriangleQ[]]]

eqn=((a^12) (b^2) (α^2)-5 (a^10) (b^4) (α^2)+10 (a^8) (b^6) (α^2)-10 (a^6) (b^8) (α^2)+5 (a^4) (b^10) (α^2)-(a^2) (b^12) (α^2)-(a^12) (c^2) (α^2)+2 (a^8) (b^4) (c^2) (α^2)+(a^6) (b^6) (c^2) (α^2)-3 (a^4) (b^8) (c^2) (α^2)+(a^2) (b^10) (c^2) (α^2)+5 (a^10) (c^4) (α^2)-2 (a^8) (b^2) (c^4) (α^2)-(a^4) (b^6) (c^4) (α^2)+(a^2) (b^8) (c^4) (α^2)-10 (a^8) (c^6) (α^2)-(a^6) (b^2) (c^6) (α^2)+(a^4) (b^4) (c^6) (α^2)+10 (a^6) (c^8) (α^2)+3 (a^4) (b^2) (c^8) (α^2)-(a^2) (b^4) (c^8) (α^2)-5 (a^4) (c^10) (α^2)-(a^2) (b^2) (c^10) (α^2)+(a^2) (c^12) (α^2)+(a^13) b α β-4 (a^11) (b^3) α β+5 (a^9) (b^5) α β-5 (a^5) (b^9) α β+4 (a^3) (b^11) α β-a (b^13) α β-2 (a^11) b (c^2) α β+9 (a^9) (b^3) (c^2) α β-17 (a^7) (b^5) (c^2) α β+17 (a^5) (b^7) (c^2) α β-9 (a^3) (b^9) (c^2) α β+2 a (b^11) (c^2) α β-2 (a^9) b (c^4) α β-(a^7) (b^3) (c^4) α β+(a^3) (b^7) (c^4) α β+2 a (b^9) (c^4) α β+8 (a^7) b (c^6) α β-2 (a^5) (b^3) (c^6) α β+2 (a^3) (b^5) (c^6) α β-8 a (b^7) (c^6) α β-7 (a^5) b (c^8) α β+7 a (b^5) (c^8) α β+2 (a^3) b (c^10) α β-2 a (b^3) (c^10) α β+(a^12) (b^2) (β^2)-5 (a^10) (b^4) (β^2)+10 (a^8) (b^6) (β^2)-10 (a^6) (b^8) (β^2)+5 (a^4) (b^10) (β^2)-(a^2) (b^12) (β^2)-(a^10) (b^2) (c^2) (β^2)+3 (a^8) (b^4) (c^2) (β^2)-(a^6) (b^6) (c^2) (β^2)-2 (a^4) (b^8) (c^2) (β^2)+(b^12) (c^2) (β^2)-(a^8) (b^2) (c^4) (β^2)+(a^6) (b^4) (c^4) (β^2)+2 (a^2) (b^8) (c^4) (β^2)-5 (b^10) (c^4) (β^2)-(a^4) (b^4) (c^6) (β^2)+(a^2) (b^6) (c^6) (β^2)+10 (b^8) (c^6) (β^2)+(a^4) (b^2) (c^8) (β^2)-3 (a^2) (b^4) (c^8) (β^2)-10 (b^6) (c^8) (β^2)+(a^2) (b^2) (c^10) (β^2)+5 (b^4) (c^10) (β^2)-(b^2) (c^12) (β^2)-(a^13) c α γ+2 (a^11) (b^2) c α γ+2 (a^9) (b^4) c α γ-8 (a^7) (b^6) c α γ+7 (a^5) (b^8) c α γ-2 (a^3) (b^10) c α γ+4 (a^11) (c^3) α γ-9 (a^9) (b^2) (c^3) α γ+(a^7) (b^4) (c^3) α γ+2 (a^5) (b^6) (c^3) α γ+2 a (b^10) (c^3) α γ-5 (a^9) (c^5) α γ+17 (a^7) (b^2) (c^5) α γ-2 (a^3) (b^6) (c^5) α γ-7 a (b^8) (c^5) α γ-17 (a^5) (b^2) (c^7) α γ-(a^3) (b^4) (c^7) α γ+8 a (b^6) (c^7) α γ+5 (a^5) (c^9) α γ+9 (a^3) (b^2) (c^9) α γ-2 a (b^4) (c^9) α γ-4 (a^3) (c^11) α γ-2 a (b^2) (c^11) α γ+a (c^13) α γ+2 (a^10) (b^3) c β γ-7 (a^8) (b^5) c β γ+8 (a^6) (b^7) c β γ-2 (a^4) (b^9) c β γ-2 (a^2) (b^11) c β γ+(b^13) c β γ-2 (a^10) b (c^3) β γ-2 (a^6) (b^5) (c^3) β γ-(a^4) (b^7) (c^3) β γ+9 (a^2) (b^9) (c^3) β γ-4 (b^11) (c^3) β γ+7 (a^8) b (c^5) β γ+2 (a^6) (b^3) (c^5) β γ-17 (a^2) (b^7) (c^5) β γ+5 (b^9) (c^5) β γ-8 (a^6) b (c^7) β γ+(a^4) (b^3) (c^7) β γ+17 (a^2) (b^5) (c^7) β γ+2 (a^4) b (c^9) β γ-9 (a^2) (b^3) (c^9) β γ-5 (b^5) (c^9) β γ+2 (a^2) b (c^11) β γ+4 (b^3) (c^11) β γ-b (c^13) β γ-(a^12) (c^2) (γ^2)+(a^10) (b^2) (c^2) (γ^2)+(a^8) (b^4) (c^2) (γ^2)-(a^4) (b^8) (c^2) (γ^2)-(a^2) (b^10) (c^2) (γ^2)+(b^12) (c^2) (γ^2)+5 (a^10) (c^4) (γ^2)-3 (a^8) (b^2) (c^4) (γ^2)-(a^6) (b^4) (c^4) (γ^2)+(a^4) (b^6) (c^4) (γ^2)+3 (a^2) (b^8) (c^4) (γ^2)-5 (b^10) (c^4) (γ^2)-10 (a^8) (c^6) (γ^2)+(a^6) (b^2) (c^6) (γ^2)-(a^2) (b^6) (c^6) (γ^2)+10 (b^8) (c^6) (γ^2)+10 (a^6) (c^8) (γ^2)+2 (a^4) (b^2) (c^8) (γ^2)-2 (a^2) (b^4) (c^8) (γ^2)-10 (b^6) (c^8) (γ^2)-5 (a^4) (c^10) (γ^2)+5 (b^4) (c^10) (γ^2)+(a^2) (c^12) (γ^2)-(b^2) (c^12) (γ^2));

Factor[CircleFunction[eqn]]

Factor[Together//@DistanceTrilinears[TrigExpand[CircleFunction[LesterCircle]]]]

Center

CenterName[X[1116]]

CenterFunction[X[1116]]

Symmetrize[Factor[CenterFunction[X[1116]]]]

IncidentCenters[KimberlingCenters,CircleCenter[LesterCircle]]

Symmetrize[Factor[Together//@DistanceTrilinears[ExactTrilinears[CircleCenter[LesterCircle]][[1]]/.Cos[2x_]:>TrigExpand[Cos[2x]]]]]

Symmetrize[CircleCenter[LesterCircle][[1]]]

Radius

Timing[eqn=Simplify[Together//@DistanceTrilinears[CircleRadius[LesterCircle]/.Cos[2x_]:>TrigExpand[Cos[2x]]],
a>0&&b>0&&c>0&&a+b>c&&b+c>a&&a+c>b]]

Derivation

rorig=Factor[Together//@DistanceTrilinears[Distance[X[3],X[1116]]]]

FullSimplify[Together//@DistanceTrilinears[Distance[X[3],X[1116]]],TriangleQ[]]

Pull out multiplicative factor

1/6/Factor[FullSimplify[Sqrt[(((((a^2)-(b^2)))^2) (((a-b-c))^2) (((a+b-c))^2) (((a-b+c))^2) (((a+b+c))^2) ((((a^2)-(c^2)))^2) ((((b^2)-(c^2)))^2))],TriangleQ[]]Circumradius[]^2/R^2]

Factor[Expand[((((((a^2)-(b^2)))^2) ((a^2)+(b^2))-((a^4)-3 (a^2) (b^2)+(b^4)) (c^2)-((a^2)+(b^2)) (c^4)+(c^6)) (((((a^2)-(b^2)))^6) ((a^4)+(a^2) (b^2)+(b^4))-((((a^2)-(b^2)))^4) (5 (a^6)+(a^4) (b^2)+(a^2) (b^4)+5 (b^6)) (c^2)+(10 (a^12)-27 (a^10) (b^2)+33 (a^8) (b^4)-23 (a^6) (b^6)+33 (a^4) (b^8)-27 (a^2) (b^10)+10 (b^12)) (c^4)-((a^2)+(b^2)) (11 (a^8)-24 (a^6) (b^2)+47 (a^4) (b^4)-24 (a^2) (b^6)+11 (b^8)) (c^6)+(10 (a^8)+13 (a^6) (b^2)+33 (a^4) (b^4)+13 (a^2) (b^6)+10 (b^8)) (c^8)-((a^2)+(b^2)) (11 (a^4)+16 (a^2) (b^2)+11 (b^4)) (c^10)+(10 (a^4)+19 (a^2) (b^2)+10 (b^4)) (c^12)-5 ((a^2)+(b^2)) (c^14)+(c^16)))]]

CircleRadius[LesterCircle]

Incident Points

(ctrs=IncidentCenters[KimberlingCenters,LesterCircle])//Timing

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

Orthogonality

Check

Timing[expr=Together//@DistanceTrilinears[First[OrthogonalQ[LesterCircle,OrthocentroidalCircle]]/.Cos[x_]:>TrigExpand[Cos[x]]]]

Graphic

t=Triangle[{{0,0},{1,0},{.1,.7}}];
circs={LesterCircle,OrthocentroidalCircle};
centers={"(Subscript[X, 1116])","(Subscript[X, 381])"};
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->.04
]
}],AspectRatio->Automatic,PlotRange->All]