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Topic: polygons thrugh a given set of points
Replies: 63   Last Post: Jun 14, 2014 4:29 AM

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 Marnie Northington Posts: 1,251 Registered: 12/13/04
Re: polygons thrugh a given set of points
Posted: Jun 12, 2014 1:51 AM

quasi <quasi@null.set> wrote:
> Let n be an integer, n >= 3. For a set of n points in the plane,
> no 3 collinear, is there always at least one n-gon having those
> points as vertices?

Yes. For n = 3 this certainly holds.

For n > 3, choose two points p, q in the set such that the line between
them divides the remaining n - 2 points into two non-empty sets. For
both such sets, build a smaller polygon out of {that set's elements} U
{p, q} by induction, then join the two such polygons by equating the
mutual pq edge. Since no three points are collinear, no two remaining
edges are collinear, so it is easily seen that there are, in fact, n
edges.

This seems too straight-forward. Have I missed a catch?

Date Subject Author
6/12/14 quasi
6/12/14 Marnie Northington
6/12/14 quasi
6/12/14 William Elliot
6/12/14 quasi
6/12/14 Marnie Northington
6/12/14 quasi
6/12/14 Marnie Northington
6/12/14 quasi
6/12/14 Marnie Northington
6/12/14 quasi
6/12/14 Timothy Murphy
6/12/14 quasi
6/12/14 Peter Percival
6/12/14 quasi
6/13/14 Peter Percival
6/13/14 Port563
6/13/14 Peter Percival
6/13/14 Port563
6/12/14 Port563
6/12/14 quasi
6/12/14 Port563
6/12/14 quasi
6/12/14 quasi
6/13/14 Port563
6/13/14 quasi
6/13/14 Port563
6/13/14 quasi
6/13/14 Port563
6/13/14 scattered
6/12/14 quasi
6/13/14 Port563
6/12/14 David Hartley
6/12/14 quasi
6/12/14 quasi
6/13/14 Port563
6/13/14 David Hartley
6/13/14 Peter Percival
6/13/14 Port563
6/13/14 quasi
6/13/14 Port563
6/13/14 quasi
6/13/14 quasi
6/13/14 Port563
6/13/14 Peter Percival
6/13/14 quasi
6/12/14 Bart Goddard
6/12/14 scattered
6/12/14 scattered
6/12/14 quasi
6/12/14 David Hartley
6/12/14 quasi
6/12/14 Richard Tobin
6/12/14 quasi
6/13/14 William Elliot
6/13/14 quasi
6/13/14 William Elliot
6/13/14 quasi
6/13/14 David Hartley
6/13/14 quasi
6/13/14 Phil Carmody
6/13/14 Phil Carmody
6/14/14 quasi
6/14/14 quasi