"David Bernier" <email@example.com> wrote in message news:firstname.lastname@example.org... > 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 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.