Triangle Dissection

Author

Eric W. Weisstein
October 25, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Dissections/TriangleDissection.nb.

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

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

Code

label[x_]:=Module[{dd=FullSimplify@Function[r,Sqrt[r.r]][Subtract@@First[x]]},
If[IntegerQ[dd],dd,TraditionalForm[HoldForm[d]]]
]

TriangleDissection[a_,p__List]:=
Module[{ts=Triangle[a,Sequence@@#]&/@{p},
s},
s=Flatten[{Line[Last/@Vertices/@ts],Sides/@ts}];
{#,Text[label[#],Coordinates[Midpoint[#]],Background->White]}&/@s
]

Analytic

Reduce[
x1^2+y1^2==e^2&&
(x1-a)^2+y^2==f^2&&
(x2-x1)^2+(y2-y1)^2==d^2&&x2^2+y2^2==b^2&&
(x2-a)^2+y2^2==c^2&&(a|b|c|d|e|f)∈Integers&&a>0&&b>0&&c>0&&d>0&&e>0&&f>0,{a,b,c,d,e,f}]

With[{z=10},
Reduce[
x1^2+y1^2==e^2&&
(x1-a)^2+y^2==f^2&&
(x2-x1)^2+(y2-y1)^2==d^2&&x2^2+y2^2==b^2&&
(x2-a)^2+y2^2==c^2&&(a|b|c|d|e|f)∈Integers&&Sequence@@(0<#<z&/@{a,b,c,d,e,f}),{a,b,c,d,e,f}]
]

Figure

Show[Graphics[TriangleDissection[22,{19,8},{17,6}]],AspectRatio->Automatic,PlotRange->All]

Almost

{x1,y1}=Select[{x,y}/.Solve[{(x^2+y^2)==27^2,(x-30)^2+y^2==22^2},{x,y}],Positive[Last[#]]&][[1]]

{x2,y2}=Select[{x,y}/.Solve[{(x^2+y^2)==23^2,(x-30)^2+y^2==16^2},{x,y}],Positive[Last[#]]&][[1]]

d=Simplify[Sqrt[(x1-x2)^2+(y1-y2)^2]]

N[d-7,10]

Show[Graphics[TriangleDissection[30,{23,16},{27,22}]],AspectRatio->Automatic]