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

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LudovicoVan

Posts: 3,201
From: London
Registered: 2/8/08
Re: Collinear
Posted: May 7, 2013 11:46 AM
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"David Bernier" <david250@videotron.ca> wrote in message
news:kmad1m$qfd$1@dont-email.me...
> 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.


An attempt:

For the base case with 3 points: if any line L with the first 2 points on it
has the 3rd point too on it, then by definition it follows that the 3 points
are collinear.

For the successor case: if we add an (n+1)th point to the collection with n
points, such that it is collinear to any 2 of the n points at the previous
step, then from the assumption that the n points were all collinear follows,
by transitivity, that the (n+1)th point too must be collinear to all other n
points.

Julio





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