McCay Circles

Author

Eric W. Weisstein
December 21, 2004

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

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

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

Graphic

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{Blue,McCayCircles[t]},
{Green,m=BrocardTriangle[2][t]},
{Red,BrocardCircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[m],{"(Subscript[M, A])","(Subscript[M, B])","(Subscript[M, C])"}}]
],LabelOffset->.03]
}],AspectRatio->Automatic,PlotRange->All]

McCayCircles[tri_List]:=Module[{
b2=Drop[BrocardTriangle[2,tri][[1]],-1],
c=Centroid[tri]/.Point:>List
},
Circumcircle/@(Join[c,#]&/@Partition[b2,2,1,1])
]

tri={{0,0},{1,.1},{.7,.55}};
b2=Drop[BrocardTriangle[2,tri][[1]],-1];
c=Centroid[tri][[1]];
pts={c,"G",tri[[1]],"(Subscript[A, 1])",tri[[2]],"(Subscript[A, 2])",tri[[3]],"(Subscript[A, 3])"};
Show[Graphics[{
Triangle[tri],
{Dashing[{.02,.02}],BrocardTriangle[2,tri]},
{RGBColor[1,0,0],McKayCircles[tri]},
PointLabels[pts,.04,.04]
},AspectRatio->Automatic]]

McCayAnimation[w_,{v2_,v3_},st_,opts___]:=Module[{n=NeubergCircle[w,{v2,v3}],v1,b,g},
Table[Show[Graphics[{
PointSize[.04],
Point[b=BrocardPoint[1,{v1=n[[1]]+n[[2]]{Cos[t],Sin[t]},v2,v3}][[1]]],
g=Centroid[{v1,v2,v3}];
PointLabels[{b,"Ω",v1,"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[A, 3])"},.04,.04],
BrocardAngles[1,{v1,v2,v3}],
AngleBisectorLabel[{"ω",.2},{b,v3,v1}],
AngleBisectorLabel[{"ω",.2},{b,v1,v2}],
AngleBisectorLabel[{"ω",.2},{b,v2,v3}],
Triangle[{v1,v2,v3}],
BrocardTriangle[2,{v1,v2,v3}],
{Thickness[.02],Line[{v2,v3}]},
Point/@{v2,v3},
{RGBColor[1,0,0],n,Point[v1]},
{RGBColor[0,0,1],
McKayCircles[{v1,v2,v3}][[2]],
PointLabels[{g,"G"},.04,.04]
}
},AspectRatio->Automatic],opts,PlotRange->All],
{t,0,2Pi-2Pi/st,2Pi/st}]
]

tri={{0,0},{1,0},{.8,.6}};
ω=.5;
n=NeubergCenters[ω,tri];

Export[gifs<>"mccay.gif",McCayAnimation[.5,{{0,0},{1,0}},20,PlotRange->{{-.1,1.1},{-.1,1.3}}]]

Show[GraphicsArray[Partition[McCayAnimation[.5,{{0,0},{1,0}},6,PlotRange->{{-.1,1.1},{-.1,1.3}},
DisplayFunction->Identity],3]]]

Derivation

Factor[Together//@ToSidelengths[TrilinearEquation@@@(Append[#,Trilinears[Centroid]]&/@Partition[TrilinearVertexMatrix[BrocardTriangle[2]],2,1,1])]]

eqns=Symmetrize/@Numerator[%]

ToAngles[-3b c  Factor[CircleFunction[eqns[[2]]]]]

Simplify[Factor[Together//@CircleRadius[eqns[[2]]]],TriangleQ[]]

Simplify[Factor[Together//@CircleRadius[eqns[[2]]] (R/(Circumradius[]))],TriangleQ[]]

Map[Factor,Simplify[Together//@((1/6)c (Sqrt[Cot[BrocardAngle[]]^2-3])),TriangleQ[]],{2}]

InputForm[(1/6)c (Sqrt[Cot[w]^2-3])]