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

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David Bernier

Posts: 3,892
Registered: 12/13/04
Re: Collinear
Posted: May 7, 2013 4:16 AM
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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) ...


Jesus is an Anarchist. -- J.R.

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