Second Brocard Triangle

Author

Eric W. Weisstein
November 11, 2005

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

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

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

Figures

t=Triangle[{{0,0},{0.7,0.71},{1.2,-.05}}];
v=Vertices[t];
b1=Coordinates@BrocardPoint[1][t];
b2=Coordinates@BrocardPoint[2][t];
sym=SymmedianPoint[t];
bt2=Vertices[BrocardTriangle[2][t]];

Show[Graphics[{
t,
{
Orange,
BrocardCircle[t]
},
{
Black,
Thickness[.007],
Triangle[bt2]
},
{
Red,
Circumcircle[Triangle@{v[[1]],v[[3]],b1}],
Circumcircle[Triangle@{v[[1]],v[[2]],b2}]
},
{
Green,
Circumcircle[Triangle@{v[[2]],v[[3]],b2}],
Circumcircle[Triangle@{v[[1]],v[[2]],b1}]
},
{
Blue,
Circumcircle[Triangle@{v[[2]],v[[3]],b1}],
Circumcircle[Triangle@{v[[1]],v[[3]],b2}]
},
{Table[{{Red,Green,Blue}[[i]],LongestLine[{v[[i]],Coordinates@sym,bt2[[i]]}]},{i,3}]},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[A, 3])"}}],
Transpose[{Point/@bt2,{"(Subscript[C, 1])","(Subscript[C, 2])","(Subscript[C, 3])"},{Red,Green,Blue}}],
Transpose[{BrocardPoint[#][t]&/@{1,2},{"Ω","Ω'"},{Red,Blue}}],
{{SymmedianPoint[t],"K",Purple}}
],
{PointSize[.02],LabelOffset->.04}
]
}],AspectRatio->Automatic,PlotRange->{{-.1,1.4},{-.2,.8}}
]

Concurrence at Symmedian Point

t=Triangle[{{0,0},{0.7,0.71},{1.2,-.05}}];
v=Vertices[t];
b1=Coordinates@BrocardPoint[1][t];
b2=Coordinates@BrocardPoint[2][t];
sym=SymmedianPoint[t];
bt2=Vertices[BrocardTriangle[2][t]];

Show[Graphics[{
t,
{Orange,BrocardCircle[t]},
{Black,Thickness[.007],
Triangle[bt2]
},
{Table[{{Red,Green,Blue}[[i]],LongestLine[{v[[i]],Coordinates@sym,bt2[[i]]}]},{i,3}]},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[A, 3])"}}],
Transpose[{Point/@bt2,{"(Subscript[C, 1])","(Subscript[C, 2])","(Subscript[C, 3])"},{Red,Green,Blue}}],
Transpose[{BrocardPoint[#][t]&/@{1,2},{"Ω","Ω'"},{Red,Blue}}],
{{SymmedianPoint[t],"K",Purple}}
],
{PointSize[.025],LabelOffset->{-.02,-.05}}
]
}],AspectRatio->Automatic,PlotRange->All
]

Centroid

Show[Graphics[{
t,
{Red,b1=BrocardTriangle[1][t]},
{Blue,b2=BrocardTriangle[2][t]},
{Green,lines},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[A, 3])"}}],
Append[#,Red]&/@Transpose[{Point/@Vertices[b1],{"(Subscript[B, 1])","(Subscript[B, 2])","(Subscript[B, 3])"}}],
Append[#,Blue]&/@Transpose[{Point/@Vertices[b2],{"(Subscript[C, 1])","(Subscript[C, 2])","(Subscript[C, 3])"}}],
{{Centroid[t],"G",Green}}
],
{PointSize[.025],LabelOffset->.03}
]
}],AspectRatio->Automatic,PlotRange->All]

t=Triangle[{{0,0},{1,0},{.2,.5}}];
Show[Graphics[{
t,
{Red,b1=BrocardTriangle[1][t]},
{Blue,b2=BrocardTriangle[2][t]},
{Green,lines},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[A, 3])"}}],
Append[#,Red]&/@Transpose[{Point/@Vertices[b1],{"(Subscript[B, 1])","(Subscript[B, 2])","(Subscript[B, 3])"}}],
Append[#,Blue]&/@Transpose[{Point/@Vertices[b2],{"(Subscript[C, 1])","(Subscript[C, 2])","(Subscript[C, 3])"}}],
{{Centroid[t],"G",Green}}
],
{PointSize[.025],LabelOffset->.03}
]
}],AspectRatio->Automatic,PlotRange->All]

persp=Factor[Simplify[Together//@(Perspector@@(BrocardTriangle/@{1,2})),TriangleQ[]]]

Symmetrize/@(-(a^2+b^2)ToSidelengths[persp])

Trilinear Vertex Matrix

TrilinearVertexMatrix[BrocardTriangle[2]]//TraditionalForm

Area

Factor[Together//@ToSidelengths[Area[BrocardTriangle[2]]](Δ/(Area[]))]

Side Lengths

Simplify[Map[Factor,Factor[Together//@ToSidelengths[SideLengths[BrocardTriangle[2]]]],{3}],TriangleQ[]]

Map[Factor,%51,{3}]

{(Sqrt[(((a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4))/((2 (a^2)+2 (b^2)-(c^2)) (2 (a^2)-(b^2)+2 (c^2))))])a,(Sqrt[(((a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4))/((-(a^2)+2 (b^2)+2 (c^2)) (2 (a^2)+2 (b^2)-(c^2))))])b,(Sqrt[(((a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4))/((-(a^2)+2 (b^2)+2 (c^2)) (2 (a^2)-(b^2)+2 (c^2))))])c}

Brocard Circle Is the Circumcircle

Factor[Together//@ToSidelengths[TrilinearEquation@@TrilinearVertexMatrix[BrocardTriangle[2]]]]

Symmetrize[-Numerator[%]]

-Numerator[Factor[TrilinearEquation[BrocardCircle]]]

Derivation

TrilinearEquation[Line[#,Trilinears[SymmedianPoint]]]&/@IdentityMatrix[3]

m1=Factor[Refine[FullSimplify[((4 (-(a^4)+(a^2) (b^2)+2 (b^4)+(a^2) (c^2)+4 (b^2) (c^2)+2 (c^4)))/(Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]))Together//@({α,β,γ}/.Solve[{TrilinearEquation[BrocardCircle]==0,-c β+b γ==0,a α+b β+c γ==2Area[]},{α,β,γ}])],TriangleQ[]]]

m2=Factor[Refine[FullSimplify[((4 ((a^2)+(b^2)+(c^2)) (2 (a^2)-(b^2)+2 (c^2)))/(Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]))Together//@({α,β,γ}/.Solve[{TrilinearEquation[BrocardCircle]==0,c α-a γ==0,a α+b β+c γ==2Area[]},{α,β,γ}])],TriangleQ[]]]

m3=Factor[Refine[FullSimplify[((4 (2 (a^2)+2 (b^2)-(c^2)) ((a^2)+(b^2)+(c^2)))/(Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]))Together//@({α,β,γ}/.Solve[{TrilinearEquation[BrocardCircle]==0,-b α+a β==0,a α+b β+c γ==2Area[]},{α,β,γ}])],TriangleQ[]]]

First[Transpose[{m1,m2,m3}]]

Case 1

t

#/First[#]&/@Flatten[Transpose[{m1,m2,m3}],1]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

#/First[#]&[Trilinears[SymmedianPoint]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]]

Case 2

t=Triangle[{{0,0},{-.2,.3},{1,0}}];

#/First[#]&/@Flatten[Transpose[{m1,m2,m3}],1]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

#/First[#]&[Trilinears[SymmedianPoint]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]]

ms[[{2,4,6}]]

{{a ,-(-(a^4)-2 (a^2) (b^2)+2 (b^4)-2 (a^2) (c^2)-2 (b^2) (c^2)-(c^4)-((a^2)+(c^2)) Abs[(a^2)-2 (b^2)+(c^2)])/(b(3 (a^2)+3 (c^2)-Abs[(a^2)-2 (b^2)+(c^2)])),c },
{-(2 (a^4)-2 (a^2) (b^2)-(b^4)-2 (a^2) (c^2)-2 (b^2) (c^2)-(c^4)+((b^2)+(c^2)) Abs[-2 (a^2)+(b^2)+(c^2)])/(a(3 (b^2)+3 (c^2)+Abs[-2 (a^2)+(b^2)+(c^2)])),b ,c },
{a ,b,((a^4)+2 (a^2) (b^2)+(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-2 (c^4)-((a^2)+(b^2)) Abs[(a^2)+(b^2)-2 (c^2)])/(c(3 (a^2)+3 (b^2)+Abs[(a^2)+(b^2)-2 (c^2)]))}}//InputForm

expr={{a (3 (a^2)+3 (c^2)-Abs[(a^2)-2 (b^2)+(c^2)]),-(1/b)(-(a^4)-2 (a^2) (b^2)+2 (b^4)-2 (a^2) (c^2)-2 (b^2) (c^2)-(c^4)-(a^2) Abs[(a^2)-2 (b^2)+(c^2)]-(c^2) Abs[(a^2)-2 (b^2)+(c^2)]),c (3 (a^2)+3 (c^2)-Abs[(a^2)-2 (b^2)+(c^2)])},{-(1/a)(2 (a^4)-2 (a^2) (b^2)-(b^4)-2 (a^2) (c^2)-2 (b^2) (c^2)-(c^4)+(b^2) Abs[-2 (a^2)+(b^2)+(c^2)]+(c^2) Abs[-2 (a^2)+(b^2)+(c^2)]),b (3 (b^2)+3 (c^2)+Abs[-2 (a^2)+(b^2)+(c^2)]),c (3 (b^2)+3 (c^2)+Abs[-2 (a^2)+(b^2)+(c^2)])},{a (3 (a^2)+3 (b^2)+Abs[(a^2)+(b^2)-2 (c^2)]),b (3 (a^2)+3 (b^2)+Abs[(a^2)+(b^2)-2 (c^2)]),(1/c)((a^4)+2 (a^2) (b^2)+(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-2 (c^4)-(a^2) Abs[(a^2)+(b^2)-2 (c^2)]-(b^2) Abs[(a^2)+(b^2)-2 (c^2)])}};

FullSimplify[expr,TriangleQ[]]

FullSimplify[expr,TriangleQ[],ExcludedForms->{Abs[x_]}]

Developer`SetSystemOptions[SimplificationOptions->SimplifyToPiecewise->False]

Simplify[expr,TriangleQ[]]

FullSimplify[expr,TriangleQ[]]

ToAngles[Factor[{{-((2*a^4-2*a^2*b^2-b^4-2*a^2*c^2-2*b^2*c^2-c^4+(b^2+c^2)*abs[-2*a^2+b^2+c^2])/(a*(3*b^2+3*c^2+abs[-2*a^2+b^2+c^2]))),b,c},{a,-((-a^4-2*a^2*b^2+2*b^4-2*a^2*c^2-2*b^2*c^2-c^4+(a^2+c^2)*abs[a^2-2*b^2+c^2])/(b*(3*a^2+3*c^2+abs[a^2-2*b^2+c^2]))),c},{a,b,(a^4+2*a^2*b^2+b^4+2*a^2*c^2+2*b^2*c^2-2*c^4+(a^2+b^2)*abs[a^2+b^2-2*c^2])/(c*(3*a^2+3*b^2+abs[a^2+b^2-2*c^2]))}
}/.abs[x_]:>-x]]{a,b,c}

Centers

(pts={#,KimberlingLookup[#[BrocardTriangle[2]]]}&/@Take[KimberlingCenters,All]);//Timing

l=Map[First,DeleteCases[pts,_?(Last[#]==={}&)],{2}]

{#[[1]],Name[#[[1]]],X[#[[2,1]]],Name[X[#[[2,1]]]]}&/@Cases[pts,_?(Last[#]=!={}&)]//TableForm