Collinear

Author

Eric W. Weisstein
January 18, 2009

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

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Trilinear

t=Triangle[{{0,0},{1,0},{.3,.5}}];

Graphics[t]

Intersections@@@Map[TrilinearLine,Subsets[{{160,200,200},{50,60,60},{150,90,90}},{2}],{2}]

Normalize/@%

Equal@@%

p=Intersections@@(TrilinearLine/@{{16,20,20},{5,6,6}})

Graphics[{
{Thickness[.005],Green,t},
Red,TrilinearLine[#][t]&/@{{16,20,20},{5,6,6},{15,9,9}},
Blue,PointSize[.02],Point[TrilinearToCartesian[t,p]]
},Axes->True,AspectRatio->Automatic,PlotRange->2,Method->{"AxesInFront"->False}]

3 Points in 3-D

Det[{{0,0,0},{1,1,1}/2,{1,1,1}}]

x1={(Subscript[x, 1]),(Subscript[y, 1]),(Subscript[z, 1])};
x2={(Subscript[x, 2]),(Subscript[y, 2]),(Subscript[z, 2])};
x3=v1+c(x2-x1);

Transpose[{x1,x2,x3}]//TraditionalForm

#.#&[Cross[x2-x1,x1-x3]]

Det[{x1,x2,x3}]

det==0

Det[{x1,x2,x3}={{(Subscript[x, 1]),(Subscript[y, 1]),(Subscript[z, 1])},{(Subscript[x, 2]),(Subscript[y, 2]),(Subscript[z, 2])},{0,0,0}}]

det==0 is necessary but not sufficient

Det[{x1,x2,x3}={{16,20,20},{5,6,6},{15,9,9}}]

#.#&[Cross[x2-x1,x1-x3]]

Graphics3D[{PointSize[.02],Point[{x1,x2,x3}]}]