Brocard Circle

Author

Eric W. Weisstein
December 21, 2004

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

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

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

Diagram

t=Triangle[{{0.,0.},{1.3,1.7},{3.,0.}}];
Show[Graphics[{
t,
BrocardCircle[t],
{
CornflowerBlue,
Line[Coordinates/@{SymmedianPoint[t],Circumcenter[t]}]
},
{
Green,
Circumcircle[t]
},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
Transpose[{BrocardPoint[#][t]&/@{1,2},{"Ω","Ω'"},{Red,Blue}}],
{
{BrocardPoint[1][t],"Ω",Red},
{BrocardPoint[2][t],"Ω'",Blue},
{SymmedianPoint[t],"K",Purple},
{Circumcenter[t],"O",Green},
{X[182][t],"(Subscript[X, 182])",CornflowerBlue}
}
],
{PointSize[.03],LabelOffset->.1}
]
}],
 AspectRatio->Automatic,PlotRange->All]

CenterName[X[#]]&/@{3,6}

Circle Function

eqn=a b c(α^2+β^2+γ^2)-a^3 β γ-b^3 γ α-c^3 α β;

CircleFunction[eqn]

IncidentCenters[KimberlingCenters,CyclicTrilinears[CircleFunction[BrocardCircle]]]

CenterName[X[2]]

Center

Together//@DistanceTrilinears[CircleCenter[BrocardCircle]]

IncidentCenters[KimberlingCenters,%]

CenterFunction[X[182]]

CenterName[X[182]]

Radius

R=FullSimplify[Together//@DistanceTrilinears[CircleRadius[BrocardCircle]],TriangleQ[]]

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

FullSimplify[R/(Circumradius[]Sqrt[1-4Sin[BrocardAngle[]]^2]/Cos[BrocardAngle[]]/2),TriangleQ[]]

Incident Centers

IncidentCenters[KimberlingCenters,CircleEquation[BrocardCircle]]//Timing

CenterName[X[1083]]

CenterFunction[X[1083]]

Orthogonality

Check

Together//@DistanceTrilinears[First[OrthogonalQ[BrocardCircle,ParryCircle]]]//Timing

Graphic

t=Triangle[{{0,0},{1,0},{.1,.7}}];
circs={BrocardCircle,ParryCircle};
centers={"(Subscript[X, 182])","(Subscript[X, 351])"};
colors={Red,Green};
Show[Graphics[{
Transpose[{colors,Through[circs[t]]}],
PointLabels[Join[
Transpose[{CircleCenter[#[t]]&/@circs,centers,colors}]
],LabelOffset->.1
]
}],AspectRatio->Automatic,PlotRange->All]

Derivation

eqn=Symmetrize[Numerator[Factor[Together//@TrilinearEquation@@TrilinearVertexMatrix[BrocardTriangle[1]]]]]

Numerator[-Factor[TrilinearEquation[BrocardCircle]]]

Factor[CircleFunction[eqn]]

Symmetrize/@Center[eqn]

Simplify[Together//@CircleRadius[eqn],TriangleQ[]]

Map[Factor,%,{2}]