de Longchamps Ellipse

Author

Eric W. Weisstein
January 29, 2005

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

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

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

Equation

5-Point Conic Equation

pts1=TrilinearVertexMatrix[IncentralTriangle]

pts2=CevianTriangle[X[57]]

Simplify[Conic[Join[pts1,Most[pts2]]]]

eqn=a (α+β-γ) (α-β+γ)+b (α+β-γ)(-α+β+γ)+c (α-β+γ)(-α+β+γ);

Coefficient[eqn,Sequence@@#]&/@{{α,2},{β,2},{γ,2},{α β,1},{α γ,1},{β γ,1}}

(a-b-c)(α^2)+(-a+b-c)(β^2)+(-a-b+c)(γ^2)+2(a β γ+b α γ+c α β)

eqn2=(a/(-α+β+γ))+(b/(α-β+γ))+(c/(α+β-γ));

Equation

TrilinearEquation[DeLongchampsEllipse]

Conic Center

Factor[ConicCenter[ExpandAll[eqn]]]

CenterFunction[Incenter]

Figure

t=Triangle[{{0,0},{1,0},{.05,.8}}];
Show[Graphics[{
t,
{Blue,
IncentralTriangle[t],
{Dashing[{.02}],Cartesian[#[t]]&/@Cevians[Incenter]}
},
{Red,DeLongchampsEllipse[t]},
{
Purple,
Cartesian[CevianTriangle[X[57]][t]],
{Dashing[{.02}],Cartesian[#[t]]&/@Cevians[X[57]]}
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[Cartesian[CevianTriangle[Incenter][t]]],{"(Subscript[I, A])","(Subscript[I, B])","(Subscript[I, C])"}}],Append[#,Purple]&/@Transpose[{VertexPoints[Cartesian[CevianTriangle[IsogonalMittenpunkt][t]]],{"(Subscript[X, A])","(Subscript[X, B])","(Subscript[X, C])"}}],
{
{Incenter[t],"I",Blue},
{X[57][t],"(Subscript[X, 57])",Purple}
}
],
{PointSize[.02],LabelOffset->-.03}
]
}],
PlotRange->{{-.1,1.1},{-.1,.9}},AspectRatio->Automatic,ImageSize->400]

Six Point Incidence

Together//@(TrilinearEquation[DeLongchampsEllipse]/.Thread[{α,β,γ}->#])&/@Flatten[Vertices/@CevianTriangle/@{Incenter,IsogonalMittenpunkt},1]

Incident Centers

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

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

Check

Short/@(FullSimplify[DistanceTrilinears[TrigExpand[eqn/.Thread[{α,β,γ}->Trilinears[#]]]],
TriangleQ[]]&/@ctrs)//Timing

Area

d2=Together//@ToSidelengths[Distance[{α,β,γ},Trilinears[Incenter]]^2]

eqn1= ((a^2) (α^2)-a b (α^2)-a c (α^2)-(a^2) α β+2 a b α β-(b^2) α β+(c^2) α β-a b (β^2)+(b^2) (β^2)-b c (β^2)-(a^2) α γ+(b^2) α γ+2 a c α γ-(c^2) α γ+(a^2) β γ-(b^2) β γ+2 b c β γ-(c^2) β γ-a c (γ^2)-b c (γ^2)+(c^2) (γ^2));

D[eqn1,α]

eqn2=Factor[TrilinearEquation[DeLongchampsEllipse]]

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]