Casey's Theorem

Eric W. Weisstein
January 4, 2002

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

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

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

Initialization

<<PlaneGeometry`;

Solving

Solve[(1-r1-x)^2+(1-r1-y)^2==(r+r1)^2,r1]

Solve[(x-r2)^2+(1-r2-y)^2==(r+r2)^2,r2]

Solve[(x-r3)^2+(y-r3)^2==(r+r3)^2,r3]

Solve[(1-r4-x)^2+(y-r4)^2==(r+r4)^2,r4]

FullSimplify[Solve[(a-r1-r2)(a-r3-r4)+(a-r1-r4)(a-r2-r3)==Sqrt[2(a-r1-r3)^2-(r1-r3)^2]Sqrt[2(a-r2-r4)^2-(r2-r4)^2],a]]

Figure

x=.6;
y=.5;
r=.25;
r1=2+r-x-y-Sqrt[2(1+r-x)(1+r-y)];
r2=1+r+x-y-Sqrt[2(r+x)(1+r-y)];
r3=r+x+y-Sqrt[2(r+x)(r+y)];
r4=1+r-x+y-Sqrt[2(r+y)(1+r-x)];

rr1={1-r1,1-r1};
rr2={r2,1-r2};
rr3={r3,r3};
rr4={1-r4,r4};

pts={{x,y},"(x,y)",rr1,
"(1-(Subscript[r, 1]),1-(Subscript[r, 1]))",rr2,"((Subscript[r, 2]),1-(Subscript[r, 2]))",
rr3,"((Subscript[r, 3]),(Subscript[r, 3]))",rr4,"(1-(Subscript[r, 4]),(Subscript[r, 4]))"};
pts2={{x,y},"(Subscript[O, 5])",rr1,"(Subscript[O, 1])",rr2,"(Subscript[O, 2])",rr3,"(Subscript[O, 3])",rr4,"(Subscript[O, 4])"};
{t3,t1}=CircleCircleTangent[Circle[rr3,r3],Circle[rr1,r1]][[1]];

CircleCircleTangent[Circle[rr3,r3],Circle[rr1,r1]]

CaseysTheoremSquare seems to be missing

Show[GraphicsArray[{
Graphics[{
CaseysTheoremSquare[{x,y},r],
PointLabels[pts,.03,.05]
},AspectRatio->Automatic],
Graphics[{
CaseysTheoremSquare[{x,y},r],
PointLabels[pts2,.03,.05],
Line[{t3,rr3,rr1,t1,t3}],
PerpSquare[{{rr3,t3},{t3,t1}},.04,4],
PerpSquare[{{rr1,t1},{t3,t1}},.04,1],
{Dashing[{.02,.02}],
Line[{{0,r3},rr3,{r3,0}}],Line[{{0,1-r2},rr2,{r2,1}}],
Line[{{1-r1,1},rr1,{1,1-r1}}],Line[{{1,r4},rr4,{1-r4,0}}],
Line[{rr1,rr1+(t3-t1)}]
}
},AspectRatio->Automatic]
}]]