Lemoine Inellipse

Author

Eric W. Weisstein
January 29, 2005

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

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

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

Equation

Inconic parameters

Midpoint[{SymmedianPoint,Centroid}]//Together

Position[CenterFunction/@KimberlingCenters,((4 (a^2)+(b^2)+(c^2))/a),1]

KimberlingCenters[[463]]

CenterName[X[597]]

Solve[Thread[{c y+b z,a z+c x,b x+a y}=={((4 (a^2)+(b^2)+(c^2))/a),(((a^2)+4 (b^2)+(c^2))/b),(((a^2)+(b^2)+4 (c^2))/c)}],{x,y,z}]//Together//FullSimplify

{x,y,z}={((-(a^2)+2 ((b^2)+(c^2)))/(b c)),((2 (a^2)-(b^2)+2 (c^2))/(a c)),((2 ((a^2)+(b^2))-(c^2))/(a b))};

Trilinear Equations

eqn=ConicEquation[LemoineInellipse]

FullSimplify[eqn (a^2)(b^2)(c^2)]

Center

DistanceTrilinears[{c y+b z,a z+c x,b x+a y}]//Simplify

CenterFunction[X[597]]

Brianchon Point

BrianchonPoint[LemoineInellipse]

IncidentCenters[KimberlingCenters,BrianchonPoint[LemoineInellipse]]//Timing

Through[{CenterFunction,CenterName}[X[598]]]

Through[{CenterFunction,CenterName}[X[598]]]

CenterName[X[574]]

Figure

t=Triangle[{{0,0},{1,0},{.15,.8}}];
Show[Graphics[{
t,
{Red,LemoineInellipse[t]},
{Blue,
LongestLine[Coordinates/@{bp,#},1]&/@VertexPoints[t]},
{Green,LemoineTriangle[t]},
{
Red,
l=Line[Coordinates/@{SymmedianPoint[t],Centroid[t]}],
EqualSideTicksAboutMidpoint[l,1,.01]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[LemoineTriangle[t]],{"(Subscript[X, A])","(Subscript[X, B])","(Subscript[X, C])"}}],
{
{X[597][t],"(Subscript[X, 597])",{Red,LabelOffset->{.03,.05}}},
{SymmedianPoint[t],"K",{Red,LabelOffset->{-.03,.05}}},
{Centroid[t],"G",{Red,LabelOffset->{.03,.05}}}
},
{{X[597][t],"(Subscript[X, 597])",Blue}}
],
{PointSize[.025],LabelOffset->-.04}
]
}],
PlotRange->{{-.1,1.1},{-.1,.9}},AspectRatio->Automatic]

CenterName[X[597]]

Incident Centers

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

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

Area

eqn1=Numerator[Together[TrilinearEquation[LemoineInellipse]]]

eqn2=Numerator[Together[TrilinearEquation[aline=TrilinearLine[Line@@(Trilinears/@{SymmedianPoint,Centroid})]]]]

as=Together//@({α,β,γ}/.Solve[{
eqn1==0,
eqn2==0,
a α+b β+c γ==2Area//ToSidelengths
},{α,β,γ}]);

Together//@(eqn1/.Thread[{α,β,γ}->#])&/@as

ToSidelengths[Distance[Trilinears[X[597]],#]&/@as]/.Thread[{a,b,c}->{10,11,13}//N[#,10]&]

aa=Map[Factor,Refine[Sqrt[Together//@ToSidelengths[Distance[Trilinears[X[597]],as[[1]]]^2]],TriangleQ[]],3]

bline=Together//@ToSidelengths[PerpendicularLine[aline,Trilinears[BrocardMidpoint]]]

Together//@ToSidelengths[PerpendicularQ[aline,bline]]

eqn2=Factor[Numerator[Together[TrilinearEquation[bline]]]]

Circumradius//ToSidelengths

Inradius//ToSidelengths

bb=Simplify[ToSidelengths[(Δ/Area)Sqrt[Factor[aa^2-(Distance[SymmedianPoint,Centroid]/2)^2]]],TriangleQ[]]

bs=Together//@({α,β,γ}/.Solve[{
eqn1==0,
eqn2==0,
a α+b β+c γ==2Area//ToSidelengths
},{α,β,γ}]);

(2Area)^2//ToSidelengths

Factor/@GroebnerBasis[{
eqn1,
eqn2,
4(a α+b β+c γ)^2-(a+b-c) (a-b+c) (-a+b+c) (a+b+c)
},{α},{β,γ},MonomialOrder->EliminationOrder]

ToSidelengths[Distance[Trilinears[X[597]],#]&/@bs]/.Thread[{a,b,c}->{10,11,13}//N[#,10]&]

{aa,bb,Refine[π aa bb,TriangleQ[]]}