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5 points to convex quad (Posted on 2009-10-05) Difficulty: 2 of 5
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 Solution | Comment 1 of 4

Let S be the set of five points.
Let N be the number of points of
S in the boundary of the convex
hull of S.
With the restriction that no three
are collinear, N must be 3, 4, or 5.
If N=5, then pick any four for the
convex quadrilateral.
If N=4, then those four are the
convex quadrilateral.
If N=3, then we have a triangle with
two of the points inside the triangle.
Those two points determine a line.
With the restriction that no three
are collinear, that line intersects
two sides of the triangle ( does not
pass through a vertex of the triangle ).
Eliminate the vertex which is the
endpoint of the two sides. The
remaining four points of S are the
convex quadrilateral.

  Posted by Bractals on 2009-10-05 16:25:50
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