Brocard Points

Author

Eric W. Weisstein
November 26, 2004

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

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

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

Center Functions

CenterFunction[BrocardPoint[#]]&/@{1,2}

FullSimplify[BisymmetricQ[#,Signed->True]&/@BrocardPoint/@{1,2},TriangleQ[]]

Diagram

t=Triangle[{{0.,0.},{3.,0.},{1.3,1.7}}];
Show[Graphics[{
 t,
{Red,BrocardAngles[1][t]},
{Blue,BrocardAngles[2][t]},
{
Red,
{EqualAngleTicks[#,1,.3,.05,.1],Text[TraditionalForm[ω],Coordinates@EndPoint[EqualAngleTicks[#,1,.3,.12,.1][[1]]],Background->White],VertexArc[#,.3]}&/@(Prepend[#,Coordinates@BrocardPoint[1][t]]&/@Coordinates/@Sides[t])
},
{
Blue,
{EqualAngleTicks[#,2,.5,.05,.1],Text[TraditionalForm[ω'],Coordinates@EndPoint[EqualAngleTicks[#,1,.5,.13,.1][[1]]],Background->White],VertexArc[#,.5]}&/@(Append[#,Coordinates@BrocardPoint[2][t]]&/@Coordinates/@Sides[t])
},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
Transpose[{BrocardPoint[#][t]&/@{1,2},{"Ω","Ω'"},{Red,Blue}}]
],
{PointSize[.025],LabelOffset->.08}
]
}], AspectRatio->Automatic,PlotRange->All]

Rotated Labeling

t=Triangle[RotateLeft[{{0.,0.},{3.,0.},{1.3,1.7}},2]];
Show[Graphics[{
 t,
{Red,BrocardAngles[1][t]},
{Blue,BrocardAngles[2][t]},
{
Red,
{EqualAngleTicks[#,1,.3,.05,.1],Text[TraditionalForm[ω],Coordinates@EndPoint[EqualAngleTicks[#,1,.3,.12,.1][[1]]],Background->White],VertexArc[#,.3]}&/@(Prepend[#,Coordinates@BrocardPoint[1][t]]&/@Coordinates/@Sides[t])
},
{
Blue,
{EqualAngleTicks[#,2,.5,.05,.1],Text[TraditionalForm[ω'],Coordinates@EndPoint[EqualAngleTicks[#,1,.5,.13,.1][[1]]],Background->White],VertexArc[#,.5]}&/@(Append[#,Coordinates@BrocardPoint[2][t]]&/@Coordinates/@Sides[t])
},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
Transpose[{BrocardPoint[#][t]&/@{1,2},{"Ω","Ω'"},{Red,Blue}}]
],
{PointSize[.025],LabelOffset->.08}
]
}], AspectRatio->Automatic,PlotRange->All]

Flipped Labeling

t=Triangle[Reverse[{{0.,0.},{3.,0.},{1.3,1.7}}]];
Show[Graphics[{
 t,
{Red,BrocardAngles[1][t]},
{Blue,BrocardAngles[2][t]},
{
Red,
{EqualAngleTicks[#,1,.3,.05,.1],Text[TraditionalForm[ω],Coordinates@EndPoint[EqualAngleTicks[#,1,.3,.12,.1][[1]]],Background->White],VertexArc[#,.3]}&/@(Prepend[#,Coordinates@BrocardPoint[1][t]]&/@Coordinates/@Sides[t])
},
{
Blue,
{EqualAngleTicks[#,2,.5,.05,.1],Text[TraditionalForm[ω'],Coordinates@EndPoint[EqualAngleTicks[#,1,.5,.13,.1][[1]]],Background->White],VertexArc[#,.5]}&/@(Append[#,Coordinates@BrocardPoint[2][t]]&/@Coordinates/@Sides[t])
},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
Transpose[{BrocardPoint[#][t]&/@{1,2},{"Ω","Ω'"},{Red,Blue}}]
],
{PointSize[.025],LabelOffset->.08}
]
}], AspectRatio->Automatic,PlotRange->All]

Distances

Simplify[Subtract@@DistanceTrilinears[Distance[BrocardPoint[#],Circumcenter]&/@{1,2}],a>0&&b>0&&c>0]

d=FullSimplify[DistanceTrilinears[Distance[BrocardPoint[1],Circumcenter]],a>0&&b>0&&c>0]

FullSimplify[Numerator[d]Sqrt[Denominator[#]/Numerator[#]&[Denominator[d]^2]]]

FullSimplify[R(Sqrt[(((a^4)+(b^4)-(b^2) (c^2)+(c^4)-(a^2) ((b^2)+(c^2)))/((b^2) (c^2)+(a^2) ((b^2)+(c^2))))])]

All the following figures must be redone as Ω and Ω have been switched

Relationship of Brocard points to circumcircles

t=Triangle[{{0.,0.},{1.3,1.7},{3.,0.}}];
v=Vertices[t];
pts={
		BrocardPoint[1][t],Ω,BrocardPoint[2][t],Ω',
		v[[1]],A,v[[2]],B,v[[3]],C
};

Show[GraphicsArray[Table[Show[
Graphics[{
t,
PointLabels[t,pts,.04],
Table[Circumcircle[Triangle@{v[[i]],v[[j]],Coordinates@BrocardPoint[k][t]}],{i,2},{j,i+1,3}]}],
		AspectRatio->Automatic,DisplayFunction->Identity],{k,2}]]
]

Pedal Triangles

t=Triangle[{{0.,0.},{1.3,1.7},{3.,0.}}];
v=Vertices[t];
pts={
BrocardPoint[1][t],Ω,
BrocardPoint[2][t],Ω',
v[[1]],A,v[[2]],B,v[[3]],C
};

Show[GraphicsArray[Table[
Graphics[{
t,
{
Dashing[{.02,.02}],
PedalLines[t,BrocardPoint[i][t]]
},
{
Thickness[.01],PedalTriangle[t,BrocardPoint[i][t]]
},
PointLabels[t,pts]
},
  AspectRatio->Automatic],{i,2}]]
]

t={{0,0},{1.3,1.7},{3,0}};
bt1=BrocardCircumTriangle[1,t][[1]];
pts1={
		BrocardPoint[1,t],"Ω",BrocardPoint[2,t],"Ω'",
		t[[1]],"A",t[[2]],"B",t[[3]],"C",
		bt1[[2]],"A'",bt1[[3]],"B'",bt1[[1]],"C'"
};

bt2=BrocardCircumTriangle[2,t][[1]];
pts2={
		BrocardPoint[1,t],"Ω",BrocardPoint[2,t],"Ω'",
		t[[1]],"A",t[[2]],"B",t[[3]],"C",
		bt2[[2]],"C''",bt2[[3]],"A''",bt2[[1]],"B''"
};

Show[GraphicsArray[{
Show[Graphics[{
			{Thickness[.01],Triangle[t]},
			PointLabels[t,pts1,.03],
			BrocardCircumTriangle[1,t],
			Circumcircle[t],
			Dashing[{.02,.02}],Table[Line[{t[[i]],bt1[[i]]}],{i,3}]
			}],AspectRatio->Automatic,DisplayFunction->Identity],
Show[Graphics[{
			{Thickness[.01],Triangle[t]},
			PointLabels[t,pts2,.03],
			BrocardCircumTriangle[2,t],
			Circumcircle[t],
			Dashing[{.02,.02}],Table[Line[{t[[i]],bt2[[i]]}],{i,3}]
			}],AspectRatio->Automatic,DisplayFunction->Identity]
}]]

t={{0.,0.},{.8,1.2},{2.,0.}};
p=Point[LineLineIntersection[{t[[1]],BrocardPoint[1,t][[1]]},{t[[3]],LemoinePoint[t][[1]]}]];
ptsp={
		t[[1]],"A",t[[2]],"B",t[[3]],"C",
		BrocardPoint[1,t],"Ω",
		LemoinePoint,"K",Centroid,"G",
    p,"P"
};

q=Point[LineLineIntersection[{t[[1]],BrocardPoint[2,t][[1]]},{t[[3]],LemoinePoint[t][[1]]}]];
ptsq={
		t[[1]],"A",t[[2]],"B",t[[3]],"C",
		BrocardPoint[2,t],"Ω'",
		LemoinePoint,"K",Centroid,"G",
    q,"Q"
};

Show[GraphicsArray[{
Show[Graphics[{
			{Thickness[.01],Triangle[t]},
			Line[LongestLine[{t[[1]],BrocardPoint[1,t],p} /. Point->Identity]],
			Line[LongestLine[{t[[2]],Centroid[t],p}/. Point->Identity]],
			Line[LongestLine[{t[[3]],LemoinePoint[t],p}/. Point->Identity]],
  PointLabels[t,ptsp,.03,{.05,.07}]
}],AspectRatio->Automatic,PlotRange->All,DisplayFunction->Identity],
			
			Show[Graphics[{
			{Thickness[.01],Triangle[t]},
			Line[LongestLine[{t[[1]],BrocardPoint[2,t],q} /. Point->Identity]],
			Line[LongestLine[{t[[2]],Centroid[t],q}/. Point->Identity]],
			Line[LongestLine[{t[[3]],LemoinePoint[t],q}/. Point->Identity]],
  PointLabels[t,ptsq,.03,{.05,.07}]
}],AspectRatio->Automatic,PlotRange->All,DisplayFunction->Identity]
		}]]