Six Circles Theorem

Author

Eric W. Weisstein
February 22, 2004

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

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

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

Code

SixCircles::usage="SixCircles[r,triangle] gives a graphic showing the six circles theorem for the specified triangle A1A2A3 with initial circle of radius r at vertex A2."

SixCircles[r_,t_Triangle]:=
FoldList[CornerCircle[#1,t,Mod[#2-1,3]+1]&,CornerCircle[r,t],Range[2,6]]

Graphic

triangles={
Triangle[{{0,0},{1,0},{.3,.6}}],
Triangle[{{0,0},{1,0},{.2,.7}}]
};
arrangements=Transpose[{{.15,.1},triangles}];
circles=SixCircles@@#&/@arrangements;

Show[GraphicsArray[Graphics[{Last[#],{Red,SixCircles@@#}},AspectRatio->Automatic]&/@arrangements]]

Most[l_]:=Drop[l,-1]

movie=Show/@(GraphicsArray[
Table[Graphics[{triangles[[i]],{Red,#[[i]]}},
AspectRatio->Automatic,PlotRange->{{-.1,1.1},{-.2,.8}}],{i,2}],
ImageSize->400
]&/@Transpose[Reverse[NestList[Most,#,5]]&/@circles])

Animated GIF

<<Utilities`GIF`

Export[gifs<>"SixCircles.gif",movie,ConversionOptions->{"Loop"->True,"AnimationDisplayTime"->1}]//Timing