Second Brocard Circle

Author

Eric W. Weisstein
November 26, 2004

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

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

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

Derivation

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

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

Simplify[Together//@DistanceTrilinears[CircleEquation[Circumcenter,a b c √(-((a^4)+(b^4)-(b^2) (c^2)+(c^4)-(a^2) ((b^2)+(c^2)))/((a-b-c) (a+b-c) (a-b+c) (a+b+c) ((a^2) (b^2)+(a^2) (c^2)+(b^2) (c^2))))]],TriangleQ[]]

Numerator[%]

CircleFunction[%]

Factor[%]

InputForm[%]

Circle Function

CircleFunction[SecondBrocardCircle]

Symmetrize[%]

IncidentCenters[KimberlingCenters,CyclicTrilinears[%]]//Timing

CenterName[X[6]]

Circle Center

Symmetrize[Together//@DistanceTrilinears[CircleCenter[SecondBrocardCircle][[1]]]]

Symmetrize[LawOfCosines[%]]

CenterFunction[X[3]]

Circle Radius

Simplify[Together//@DistanceTrilinears[CircleRadius[SecondBrocardCircle]],TriangleQ[]]

Incident Centers

IncidentCenters[KimberlingCenters,SecondBrocardCircle]//Timing

Figure

t=Triangle[{{0,0},{1,0},{.3,.6}}];
colors={Red,Blue};
Show[Graphics[{
t,
Transpose[{colors,{BrocardCircle[t],SecondBrocardCircle[t]}}],
PointLabels[Join[
Transpose[{BrocardPoint[#][t]&/@{1,2},{"Ω","Ω'"},Purple{1,1}}],
Transpose[{VertexPoints[t],{"A","B","C"}}],
{{X[182][t],"(Subscript[X, 182])",Red},{X[3][t],"O",Blue}}
],LabelOffset->.03]
}],
AspectRatio->Automatic,PlotRange->All
]