Schoch Line

Author

Eric W. Weisstein
March 27, 2006

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

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

©2006 Wolfram Research, Inc. except for portions noted otherwise

Additional Authors

Original figure due to J.Wassenaar
November 25, 2005

Code

Arbelos[r_:0.6,o_:{0,0},s_:1]:={{GrayLevel[0.6],Disk[o+s{1/2,0},s/2,{0,Pi}]},Disk[o+s{r/2,0},s(r/2),{0,Pi}],Disk[o+s{(1+r)/2,0},s((1-r)/2),{0,Pi}],{GrayLevel[1],Disk[o+s{r/2,0},s(r/2),{0,Pi}],Disk[o+s{(1+r)/2,0},s((1-r)/2),{0,Pi}]},Circle[o+s{r/2,0},s(r/2),{0,Pi}],Circle[o+s{(1+r)/2,0},s((1-r)/2),{0,Pi}],Circle[o+s{1/2,0},s/2,{0,Pi}],Line[{o,o}+s{{0,0},{1,0}}]}

SchochCircle[r_]:=Circle[{(1/2) (r+3 (r^2)-2 (r^3)),(1-r) r (Sqrt[2+r-(r^2)])},(1/2)r(1-r)]

Derivation

Simplify[Solve[{
(Sqrt[x^2+y^2])==r+ρ,
(Sqrt[(x-1)^2+y^2])==1-r+ρ,
(Sqrt[(((x-(1/2)))^2)+(y^2)])==(1/2)-ρ},{x,y,ρ}],0<r<1]

With[{r=.6},Circle[{-(-1+r) r (Sqrt[2+r-(r^2)]),(1/2) (r+3 (r^2)-2 (r^3))},(1/2)r(1-r)]]

{-(-1+r) r (Sqrt[Factor[2+r-(r^2)]]),Factor[(1/2) (r+3 (r^2)-2 (r^3))]}

Schoch Line

With[{r=.6},
Show[Graphics[{
Arbelos[r],
 Line[{{.624,0},{.624,.6}}],
{Red,sc=SchochCircle[r]},
{Purple,Circle[{0, 0},r,{0,π}]},
{Blue,Circle[{1, 0},1-r,{0,π}]
},
PointLabels[
{{Point[{0,0}],"A",Purple},
{Point[{r,0}],"B"},
{Point[{1,0}],"C",Blue},
{Center[sc],"(Subscript[A, 1])",{Red}}
},LabelOffset->{0,-.04}],
{Purple,Text["(Subscript[K, 1])",{.3,.57}]},
{Blue,Text["(Subscript[K, 2])",{.95,.45}]},
Text["(Subscript[K, 3])",{.52,.55}]
}],AspectRatio->Automatic,PlotRange->{{-.1,1.1},{-.1,.6}}, ImageSize->300]
]