Crossed Ladders Theorem

Author

Eric W. Weisstein
September 27, 2004

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

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

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

Figure

CrossedLadders[{h1_,h2_},dd_,t_]:=Module[
{
a={0,0},
b=h1 {Cos[t],Sin[t]},
c={dd,0},
d={dd,0}+h2{Cos[t],Sin[t]},
e,f,
dist
},
e=Coordinates@Intersections[Line[{a,d}],Line[{b,c}]];
f=Coordinates@Intersections[Line[{{0,0},{1,0}}],Line[{e,e-{Cos[t],Sin[t]}}]];
pts={a,"A",b,"B",c,"C",d,"D",e,"E",f,"F"};
dist=Sqrt[#.#]&/@{a-b,c-d,e-f};
Print[{dist,{1/dist[[1]]+1/dist[[2]],1/dist[[3]]}}];
{
Line[{{0,0},h1 {Cos[t],Sin[t]},{dd,0}}],
Line[{{0,0},{dd,0}+h2{Cos[t],Sin[t]},{dd,0}}],
Line[{e,f}],
Line[{{-.1,0},{dd+.1,0}}],
PointLabels[{Point[#1],#2}&@@@Partition[pts,2],{PointSize[.03],LabelOffset->{0,-.08}}]
}
]

Show[Graphics[CrossedLadders[{1,.6},1,70.Degree],
AspectRatio->Automatic,PlotRange->All]]

Extended Crossed Ladders Theorem

Syntax needs to be updated

CrossedLadders[b_List,{i0_,h0_},t_]:=Module[
{a={0,0},
c={1,0},
h={h0,0},
i={i0,0},
d,e,f,g,j,
n={Cos[t],Sin[t]},
dist
},
d=LineLineIntersection[{h,h+n},{b,c}];
g=LineLineIntersection[{b,b-n},{a,c}];
e=LineLineIntersection[{i,i+n},{a,b}];
f=LineLineIntersection[{c,e},{d,a}];
j=LineLineIntersection[{f,f-n},{a,c}];
dist=Sqrt[#.#]&/@{e-i,d-h,f-j,b-g};
pts={a,"A",b,"B",c,"C",d,"D",e,"E",f,"F",g,"G",h,"H",i,"I",j,"J"};
Print[{Tr[1/Take[dist,2]],Tr[1/Take[dist,-2]]}];
{
Line/@{{a,c,b,a},{a,d,h},{b,g},{i,e,c},{f,j}},
PointLabels[pts,.025,{0,-.06}]
}
]

Show[Graphics[CrossedLadders[{.7,.4},{.25,.8},50Degree],
PlotRange->All,AspectRatio->Automatic]]