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Topic:
Collinear
Replies:
9
Last Post:
May 8, 2013 1:12 PM




Re: Collinear
Posted:
May 7, 2013 4:16 AM


On 05/07/2013 03:48 AM, William Elliot wrote: > Let C be a collection of n points with the property that > any line L with two points of C on it, has a third point > of C on it. > > How is it that C is collinear, ie all points of C are > on a single line? >
I think this can be proved by induction on 'n' the number of points.
For n >= 3, let P(n) denote the statement: "In any collection C of n points in the plane, if C is such that any line L with two points of C on it has a third point on it (i.e., L), then the set C consists of collinear points."
P(3) is obvious.
Then, for any n >= 3, we want to prove: P(n) implies P(n+1).
If we can do that, we're done.
Sketch: assume P(n) and let C be any collection of n+1 points in the plane. Let Q be some point in C and let C' = C \ {Q}.
Then C' has n points. By the induction hypothesis, P(n) is assumed.
xxxx I dunno ...
I would try to show that P(3) implies P(4), so try to show P(4) using the property (assumed) P(3) ...
dave
 Jesus is an Anarchist.  J.R.



