MacBeath Triangle

Author

Eric W. Weisstein
November 11, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.8}}];
bp=Cartesian[BrianchonPoint[MacBeathInconic][t]];
Show[Graphics[{
t,
{Red,MacBeathInconic[t]},
{Blue,MacBeathTriangle[t]},
{Green,LongestLine[Coordinates/@{bp,#},1]&/@VertexPoints[t]},
{
Red,
Line[Coordinates/@{Orthocenter[t],Circumcenter[t]}]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[MacBeathTriangle[t]],{"(Subscript[X, A])","(Subscript[X, B])","(Subscript[X, C])"}}],
{
{NinePointCenter[t],"N",{Red,LabelOffset->{.03,.05}}},
{Orthocenter[t],"H",{Red,LabelOffset->{-.03,.05}}},
{Circumcenter[t],"O",{Red,LabelOffset->{.03,.05}}}
},
{
{bp,"(Subscript[X, 264])",Blue}
}
],
{PointSize[.025],LabelOffset->-.04}
]
}],
PlotRange->{{-.1,1.1},{-.1,.9}},AspectRatio->Automatic]

Trilinear Vertex Matrix

Vertices[Triangle[MacBeathTriangle]]//TraditionalForm

Perspectrix

i=Intersections@@@Partition[Line@@@Transpose[TrilinearVertexMatrix/@{MacBeathTriangle,ReferenceTriangle}],2,1,1]

Map[-Symmetrize[ToAngles[Symmetrize[#]]]&,Together//@ToSidelengths[ExactTrilinears/@i],{2}]

CenterFunction[X[264]]

-Symmetrize[ToAngles[Symmetrize[Factor[ToSidelengths[TrigExpand[CenterFunction[X[264]]]]]]]]

Area

area=Factor[Together//@ToSidelengths[Area[MacBeathTriangle]]]

Simplify[ToAngles[area](Δ/(Area[])),TriangleQ[]]

Factor[Simplify[Factor[Together//@ToSidelengths[TrigExpand[2 Area[] Cos[A] Cos[B] Cos[C] Sec[A-B] Sec[A-C] Sec[B-C]]]],TriangleQ[]]]

Side Lengths

Map[Factor,Simplify[ToAngles[Map[Factor,Together//@ToSidelengths[SideLengths[MacBeathTriangle]],3]],TriangleQ[]],3]

Centers

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