Brocard Inellipse

Author

Eric W. Weisstein
December 31, 2004

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

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

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

Equation

Inconic parameters

{x,y,z}={(1/a),(1/b),(1/c)};

Trilinear Equations

eqn=ConicEquation[BrocardInellipse]

FullSimplify[eqn]

Center

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

CenterFunction[BrocardMidpoint]

Brianchon Point

BrianchonPoint[BrocardInellipse]

Trilinears[SymmedianPoint]

Figure

<<Graphics`ImplicitPlot`

t=Triangle[{{0,0},{1,0},{.3,.7}}];
bp=BrocardPoint[#][t]&/@{1,2};
bb=.267;
aa=Sqrt[bb^2-(Distance@@bp/2)^2];
eqn=Plus@@(Sqrt[(x-#1)^2+(y-#2)^2]&@@@(Coordinates/@bp))==2aa;

Show[Graphics[{
t,
{
Red,
ImplicitPlot[eqn,{x,-2,2},DisplayFunction->Identity][[1]],
Line[Coordinates/@bp]
},
{
Blue,
LongestLine[Coordinates/@{SymmedianPoint[t],#},.5]&/@VertexPoints[t]
},
{
Green,
SymmedialTriangle[t]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[SymmedialTriangle[t]],{"(Subscript[K, A])","(Subscript[K, B])","(Subscript[K, C])"}}],
{
{BrocardPoint[1][t],"Ω",Red},
{BrocardPoint[2][t],"Ω'",Red},
{BrocardMidpoint[t],"(Subscript[M, ΩΩ'])",Red},
{SymmedianPoint[t],"K",{Blue,LabelOffset->{-.035,.05}}}
}
],LabelOffset->{.05,-.02},PointSize[.025]]
}
],AspectRatio->Automatic,PlotRange->{{-.1,1.1},{-.1,.8}}]

Contact Points

c First[{α,β,γ}/.Solve[TrilinearEquation[BrocardInellipse]==0&&α==0&&γ==1,{α,β,γ}]]

c First[{α,β,γ}/.Solve[TrilinearEquation[BrocardInellipse]==0&&β==0&&γ==1,{α,β,γ}]]

b First[{α,β,γ}/.Solve[TrilinearEquation[BrocardInellipse]==0&&γ==0&&β==1,{α,β,γ}]]

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

eqn1=Numerator[Together[TrilinearEquation[BrocardInellipse]]]

eqn2=Numerator[Together[TrilinearEquation[aline=TrilinearLine[Line@@(Trilinears[BrocardPoint[#]]&/@{1,2})]]]]

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

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

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

aa=Refine[Sqrt[Together//@ToSidelengths[Distance[Trilinears[BrocardMidpoint],as[[1]]]^2]],TriangleQ[]]

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

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

Factor[Numerator[Together[TrilinearEquation[bline]]]]

eqn2=((b^3) c α-b (c^3) α-(a^3) c β+a (c^3) β+(a^3) b γ-a (b^3) γ);

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

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

bb=Map[Factor,Refine[ToSidelengths[(Δ/Area)]Sqrt[Together//@ToSidelengths[Distance[Trilinears[BrocardMidpoint],bs[[1]]]^2]],TriangleQ[]],2]

Map[Factor,Simplify[ToSidelengths[(Δ/Area)]Together//@ToSidelengths[(Sqrt[(aa^2)-(((1/2Distance@@(BrocardPoint/@{1,2})))^2)])],TriangleQ[]],3]

{aa,bb,Refine[π aa bb,TriangleQ[]]}