First Brocard Triangle

Author

Eric W. Weisstein
November 11, 2005

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

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

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

Figure

Remove[DarkGreen]

<<Graphics`Colors`

t=Triangle[{{0,0},{1,0},{.2,.5}}];
Show[Graphics[{
t,
{Purple,BrocardCircle[t]},
bt=BrocardTriangle[1][t],
{Dashing[{.02}],
Transpose[{{Red,Blue},Cartesian[#[t]]&/@Cevians[BrocardPoint[#]]&/@Range[2]}]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[A, 3])"}}],
Append[#,DarkGreen]&/@Transpose[{VertexPoints[bt],{"(Subscript[B, 1])","(Subscript[B, 2])","(Subscript[B, 3])"}}],
Transpose[{BrocardPoint[#][t]&/@{1,2},{"Ω","Ω'"},{Red,Blue}}]
],
{PointSize[.025],LabelOffset->.03}
]
}],AspectRatio->Automatic,PlotRange->All]

{Orientation[#],Angles[#],Function[f,#/Max[#]&@SideLengths[f]][#]}&/@{t,BrocardTriangle[1][t]}//ColumnForm

Diagram Showing relationship to midpoints

t=Triangle[{{0,0},{.2,.5},{1,0.}}];
v=Vertices[t];
bt=BrocardTriangle[1][t];
m=Midpoints[bt];
n=NinePointCenter[t];

Show[Graphics[{
t,
{
Red,
BrocardTriangle[1][t]
},
{
Purple,
NinePointCircle[t]
},
{
Blue,
LongestLine[Append[Coordinates[#],Coordinates@n]]&/@(PerpendicularLine@@@Transpose[{m,Sides[t]}]),
PerpSquare[#,.025,1]&/@Transpose[{Sides[t],PerpendicularLine@@@Transpose[{m,Sides[t]}]}]
},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[A, 3])"}}],
Append[#,Red]&/@Transpose[{Point/@Vertices[bt],{"(Subscript[B, 1])","(Subscript[B, 2])","(Subscript[B, 3])"}}],
Append[#,Blue]&/@Transpose[{m,{"(Subscript[M, 1])","(Subscript[M, 2])","(Subscript[M, 3])"}}],
{{n,"N",Purple}}
],
{PointSize[.02],LabelOffset->.03}
]
}],AspectRatio->Automatic,PlotRange->All]

1st Brocard Triangle Perspective to Original WRT Ω''

t=Triangle[{{0,0},{1,0},{.2,.7}}];
Show[Graphics[{
t,
{Red,BrocardTriangle[1][t]},
{Blue,Line[{Sequence@@#,Coordinates[BrocardPoint[3][t]]}]&/@Transpose[Vertices/@{t,bt=BrocardTriangle[1][t]}]},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[A, 3])"}}],
Append[#,Red]&/@Transpose[{Point/@Vertices[bt],{"(Subscript[B, 1])","(Subscript[B, 2])","(Subscript[B, 3])"}}],
{{BrocardPoint[3][t],"Ω''",Blue}}
],
{PointSize[.025],LabelOffset->.03}
]
}],AspectRatio->Automatic,PlotRange->All]

1st and 2nd Brocard Triangles Perspective WRT Centroid

t=Triangle[{{0,0},{1,0},{.2,.5}}];
lines=Line/@Transpose[Vertices/@(BrocardTriangle[#][t]&/@{1,2})]

{Intersections@@@Partition[lines,2,1,1],Centroid[t]}

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]

1st Brocard Triangle and Original Triangle Share Centroids

t=Triangle[{{0,0},{1,0},{0.4,0.6}}];
Show[Graphics[{
t,
{Red,b1=BrocardTriangle[1][t]},
{Green,Line/@Transpose[Vertices/@{t,MidpointTriangle[t]}]},
{Purple,Line/@Transpose[Vertices/@{b1,MidpointTriangle[b1]}]},
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])"}}],
{{Centroid[t],"G",Blue}}
],
{PointSize[.025],LabelOffset->.03}
]
}],AspectRatio->Automatic,PlotRange->All]

Centroid/@{t,b1}

Trilinear Vertex Matrix

TrilinearVertexMatrix[BrocardTriangle[1]]//TraditionalForm

Brocard Circle Is the Circumcircle

Factor[Together//@ToSidelengths[TrilinearEquation@@TrilinearVertexMatrix[BrocardTriangle[1]]]]

Symmetrize[Numerator[%]]

-Numerator[Factor[TrilinearEquation[BrocardCircle]]]

Area

Factor[Together//@ToSidelengths[Area[BrocardTriangle[1]]](Δ/(Area[]))]

Side Lengths

Map[Factor,Simplify[Map[Factor,Factor[Together//@ToSidelengths[SideLengths[BrocardTriangle[1]]]],{3}],TriangleQ[]],{2}]

Derivation

RotateLeft[Partition[IdentityMatrix[3],2,1,1]]

(Line@@@Transpose[{Trilinears/@BrocardPoint/@Range[2],#}]&/@RotateLeft[Partition[IdentityMatrix[3],2,1,1]])

Intersections@@@(Line@@@Transpose[{Trilinears/@BrocardPoint/@Range[2],#}]&/@RotateLeft[Partition[IdentityMatrix[3],2,1,1]])

a b c %

InputForm[%]

TrilinearVertexMatrix[BrocardTriangle[1]]

t=Triangle[{{0,0},{1,0},{.2,.5}}];
Show[Graphics[{
t,
BrocardTriangle[1][t],
{Red,Cartesian[#[t]]&/@Cevians[BrocardPoint[1]]},
{Blue,Cartesian[#[t]]&/@Cevians[BrocardPoint[2]]}
}],AspectRatio->Automatic]

Centers

(pts={#,KimberlingLookup[#[BrocardTriangle[1]]]}&/@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