Third Brocard Triangle

Author

Eric W. Weisstein
November 11, 2005

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

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

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

Graphic

t=Triangle[{{0,0},{1,0},{.3,.5}}];
Show[Graphics[{
t,
{Red,bt3=BrocardTriangle[3][t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[bt3],{"(Subscript[A, 3])","(Subscript[B, 3])","(Subscript[C, 3])"}}]
],LabelOffset->.03]
}],AspectRatio->Automatic,PlotRange->All]

Trilinear Vertex Matrix

{c b,c a,a b}^3{a, b,c}IsogonalConjugate/@TrilinearVertexMatrix[BrocardTriangle[1]]

TrilinearVertexMatrix[BrocardTriangle[3]]//TraditionalForm

Side Lengths

Factor[Together//@SideLengths[BrocardTriangle[3]]^2]

Area

Factor[Area[BrocardTriangle[3]]]

Circumcircle

cc=Circumcircle[BrocardTriangle[3]];

Radius

Factor[Simplify[Together//@CircleRadius[cc]^2]]

Center

Symmetrize[Factor[Center[cc]]]

KimberlingLookup[%]

Centers

(pts={#,KimberlingLookup[#[BrocardTriangle[3]]]}&/@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