Circle Inscribing

Eric W. Weisstein
January 11, 2002

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

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

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

Circle-Circle Triangle Sangaku Problem

CircleCircleTriangle::usage="CircleCircleTriangle[r] for 0<r<1 gives the figure for the Sangaku problem consisting of a circle and isosceles triangle enclosed in a circle and a smaller circle inscribed in the three."

CircleCircleTriangle[r_]:=Module[{a=r(1-r)/(1+r),y=r Sqrt[2(1-r)]/(1+r),h=1/2},{Circle[{0,0},h],Circle[{(r-1)/2,0},r/2],Triangle[{{r-h,0},{h,0},{r/2,Sqrt[1-r^2]/2}}],Line[{{-h,0},{h,0}}],Line[{{r-h,0},{r-h,y}}],{Dashing[{.02,.02}],Circle[{r-h,y},a]}}]

PowerExpand[FullSimplify[Solve[{(r/2+a)^2==(r/2)^2+y^2,(1/2-a)^2==(r-1/2)^2+y^2},{a,y}]]]

x={-1/2,0};
o2={(r-1)/2,0};
a={r-1/2,0};
b={1/2,0};
c={r/2,Sqrt[1-r^2]/2};
d={r/2,0};
rc={r-1/2,r Sqrt[2(1-r)]/(1+r)};
pts={x,"X",o2,"(Subscript[O, 2])",{0,0},"O",a,"A",b,"B",c,"C",d,"D",rc,"(Subscript[O, 3])"};
Show[Graphics[{
CircleCircleTriangle[r],
PointLabels[pts,.03,-.04],
{Dashing[{.02,.02}],Line[{x,rc}],Line[{c,d}],Line[{{0,0},rc/Sqrt[rc.rc]/2}]}
} /. r->.6],AspectRatio->Automatic]