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 5 points to convex quad (Posted on 2009-10-05)
Given 5 coplanar points with no three collinear, prove that there must be a subset of 4 points that form a convex quadrilateral.

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 Solution | Comment 1 of 4
`Let S be the set of five points.`
`Let N be the number of points ofS in the boundary of the convex hull of S.`
`With the restriction that no threeare collinear, N must be 3, 4, or 5.`
`If N=5, then pick any four for theconvex quadrilateral.`
`If N=4, then those four are theconvex quadrilateral.`
`If N=3, then we have a triangle withtwo of the points inside the triangle.Those two points determine a line.With the restriction that no threeare collinear, that line intersectstwo sides of the triangle ( does notpass through a vertex of the triangle ).Eliminate the vertex which is the endpoint of the two sides. The remaining four points of S are theconvex quadrilateral.`
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 Posted by Bractals on 2009-10-05 16:25:50

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