Macbeath Inconic

Author

Eric W. Weisstein
December 18, 2004

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

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

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

Equation

Inconic parameters

{x,y,z}={a^2Cos[A],b^2Cos[B],c^2Cos[C]};

Trilinear Equations

eqn=TrilinearEquation[MacbeathInconic]

FullSimplify[eqn]

coefs={α^2,β^2,γ^2,α β,β γ,α γ};

(Factor/@Coefficient[Symmetrize[ToSidelengths[eqn]],#]&/@coefs).coefs

Center

{c y+b z,a z+c x,b x+a y}

Symmetrize[ToSidelengths[(b^2) c Cos[B]+b (c^2) Cos[C]]]

Symmetrize[-ToSidelengths[Center[MacbeathInconic][[1]]]]

Symmetrize[(Together//@ToSidelengths[CenterFunction[NinePointCenter]/.Cos[x_]:>TrigExpand[Cos[x]]])/.Sqrt[x_]Sqrt[y_]:>Simplify[Sqrt[x y],TriangleQ[]]]

IncidentCenters[{NinePointCenter},Center[MacbeathInconic]]

Brianchon Point

BrianchonPoint[MacbeathInconic]

IsotomicConjugate[Circumcenter]

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

Through[{CenterName,CenterFunction}[X[264]]]

Symmetrize[LawOfCosines[Symmetrize[Together//@DistanceTrilinears[TrigExpand[Csc[A] Csc[2 A]]]]]]

Contact Points

b^2First[{α,β,γ}/.Solve[TrilinearEquation[MacbeathInconic]==0&&α==0&&γ==1,{α,β,γ}]]

a^2First[{α,β,γ}/.Solve[TrilinearEquation[MacbeathInconic]==0&&β==0&&γ==1,{α,β,γ}]]

a^2First[{α,β,γ}/.Solve[TrilinearEquation[MacbeathInconic]==0&&γ==0&&β==1,{α,β,γ}]]

Figure

FullSimplify[Solve[eqn==0/.γ->1,β],TriangleQ[]]

Off[ParametricPlot::"pptr"];
t=Triangle[{{0,0},{1,0},{.3,.8}}];
bp=Cartesian[BrianchonPoint[MacbeathInconic][t]];
Show[Graphics[{
t,
(*Cartesian[Triangle[{{0,(c^2) Cos[C] Sec[B],(b^2)},{(c^2),0,(a^2) Cos[A] Sec[C]},{(b^2) Cos[B] Sec[A],(a^2),0}}][t]],*)
{Red,
ParametricPlot[Evaluate[TrilinearToCartesian[t,{α,(1/(b^2))((((a (Sqrt[α]) (Sqrt[Cos[A]])+#c (Sqrt[Cos[C]])))^2) Sec[B]),1}/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]]/.Thread[{A,B,C}->Angles[Evaluate[t]]]],{α,-100,100},PlotRange->All,PlotPoints->100,
DisplayFunction->Identity][[1,1,1]]&/@{-1,1}
},
{
Green,
MacbeathTriangle[t]
},
{
Blue,
LongestLine[Coordinates/@{bp,#},1]&/@VertexPoints[t]
},
{
Red,
l=Line[Coordinates/@{Orthocenter[t],Circumcenter[t]}],
EqualSideTicksAboutMidpoint[l,1,.01]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[MacbeathTriangle[t]],{"(Subscript[X, A])","(Subscript[X, B])","(Subscript[X, C])"}}],
{
{NinePointCenter[t],"N",{Red,LabelOffset->{.03,.05}}},
{Orthocenter[t],"H",{Red,LabelOffset->{-.03,.05}}},
{Circumcenter[t],"O",{Red,LabelOffset->{.03,.05}}}
},
{
{bp,"(Subscript[X, 264])",Blue},
{X[339][t],"(Subscript[X, 339])",Red}
}
],
{PointSize[.025],LabelOffset->-.04}
]
}],
PlotRange->{{-.1,1.1},{-.1,.9}},AspectRatio->Automatic]

Area

Focal Distance

cc=Together//@ToSidelengths[Distance[Orthocenter,Circumcenter]/2]

Intersection of Major Axis with Ellipse

Factor[(Together//@ToSidelengths[{LineEquation[Orthocenter,Circumcenter],ConicEquation[MacbeathInconic]}])]

eqn1=((a^3) (b^2) α-a (b^4) α-(a^3) (c^2) α+a (c^4) α+(a^4) b β-(a^2) (b^3) β+(b^3) (c^2) β-b (c^4) β-(a^4) c γ+(b^4) c γ+(a^2) (c^3) γ-(b^2) (c^3) γ);
eqn2=((a^10) (α^2)-2 (a^8) (b^2) (α^2)+(a^6) (b^4) (α^2)-2 (a^8) (c^2) (α^2)+2 (a^6) (b^2) (c^2) (α^2)+(a^6) (c^4) (α^2)+2 (a^7) (b^3) α β-4 (a^5) (b^5) α β+2 (a^3) (b^7) α β-2 (a^3) (b^3) (c^4) α β+(a^4) (b^6) (β^2)-2 (a^2) (b^8) (β^2)+(b^10) (β^2)+2 (a^2) (b^6) (c^2) (β^2)-2 (b^8) (c^2) (β^2)+(b^6) (c^4) (β^2)+2 (a^7) (c^3) α γ-2 (a^3) (b^4) (c^3) α γ-4 (a^5) (c^5) α γ+2 (a^3) (c^7) α γ-2 (a^4) (b^3) (c^3) β γ+2 (b^7) (c^3) β γ-4 (b^5) (c^5) β γ+2 (b^3) (c^7) β γ+(a^4) (c^6) (γ^2)+2 (a^2) (b^2) (c^6) (γ^2)+(b^4) (c^6) (γ^2)-2 (a^2) (c^8) (γ^2)-2 (b^2) (c^8) (γ^2)+(c^10) (γ^2));

soln=Simplify[Together//@Solve[{eqn1==0,eqn2==0,γ==1},{α,β}],TriangleQ[]]

{α,β,1}/.soln[[1]]

Symmetrize/@(Together//@ExactTrilinears[{α,β,1}/.soln[[1]]])

aa=First[%];

l={ (Sqrt[(a^6)-(a^4) (b^2)-(a^2) (b^4)+(b^6)-(a^4) (c^2)+3 (a^2) (b^2) (c^2)-(b^4) (c^2)-(a^2) (c^4)-(b^2) (c^4)+(c^6)]) Abs[(b^2)-(c^2)]};

n=Factor[Coefficient[Numerator[aa],#]&/@l]

rest=Factor[Numerator[aa]-n.l]

numer=n.l+rest

d=Factor[Coefficient[Denominator[aa],#]&/@l]

rest=Factor[Denominator[aa]-d.l]

denom=d.l+rest

(numer/denom)

((a^2)-(b^2)+(c^2))(-(b-c) (b+c)  ((a^6) (b^2)-(a^4) (b^4)-(a^2) (b^6)+(b^8)+(a^6) (c^2)-(a^4) (b^2) (c^2)+3 (a^2) (b^4) (c^2)-(b^6) (c^2)-2 (a^4) (c^4)-(a^2) (b^2) (c^4)-(b^4) (c^4)+(a^2) (c^6)+(b^2) (c^6))+2 a b (a-c) c (a+c) (Sqrt[(a^6)-(a^4) (b^2)-(a^2) (b^4)+(b^6)-(a^4) (c^2)+3 (a^2) (b^2) (c^2)-(b^4) (c^2)-(a^2) (c^4)-(b^2) (c^4)+(c^6)]) Abs[(b^2)-(c^2)])/(a ((a^4)-2 (a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2))((b-c) (b+c)  ((a^6)-(a^4) (b^2)-(a^2) (b^4)+(b^6)-(a^4) (c^2)+3 (a^2) (b^2) (c^2)-(b^4) (c^2)-(a^2) (c^4)-(b^2) (c^4)+(c^6))+a b c  (Sqrt[(a^6)-(a^4) (b^2)-(a^2) (b^4)+(b^6)-(a^4) (c^2)+3 (a^2) (b^2) (c^2)-(b^4) (c^2)-(a^2) (c^4)-(b^2) (c^4)+(c^6)]) Abs[(b^2)-(c^2)]))

Off[ParametricPlot::"pptr"];
t=Triangle[{{0,0},{1,0},{.3,.8}}];
bp=Point@TrilinearToCartesian[t,{((Sec[A])/(a^2)),((Sec[B])/(b^2)),((Sec[C])/(c^2))}/.Thread[{A,B,C}->Angles[Evaluate[t]]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]];
Show[Graphics[{
t,
{Red,
ParametricPlot[Evaluate[TrilinearToCartesian[t,{α,(1/(b^2))((((a (Sqrt[α]) (Sqrt[Cos[A]])+#c (Sqrt[Cos[C]])))^2) Sec[B]),1}/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]]/.Thread[{A,B,C}->Angles[Evaluate[t]]]],{α,-100,100},PlotRange->All,PlotPoints->100,
DisplayFunction->Identity][[1,1,1]]&/@{-1,1}
},
{
Red,
Line[Coordinates/@{Orthocenter[t],Circumcenter[t]}]
},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
{
{NinePointCenter[t],"N",{Red,LabelOffset->{.03,.05}}},
{Orthocenter[t],"H",{Red,LabelOffset->{-.03,.05}}},
{Circumcenter[t],"O",{Red,LabelOffset->{.03,.05}}}
},
{
{bp,"X",Blue},
{Point@TrilinearToCartesian[t,{α,β,1}/.soln[[1]]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]],"E",Red}
}
],
{PointSize[.025],LabelOffset->-.04}
]
}],
PlotRange->{{-.1,1.1},{-.1,.9}},AspectRatio->Automatic]

Incident Centers

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

CenterName/@{X[1312],X[1313]}

CenterFunction/@{X[1312],X[1313]}

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

CenterName[X[250]]

CenterName[X[125]]