One Line through two Points

This Problem was set up by Silvester in 1893. The solution that I will show you was found by L. M. Kelly in 1948. Paul Erdős said, that this is an example from the book. Paul Erdős often spoke about a book belonging to god which notes the best proofs of every theorem.

The Problem:

Let A be a finite amount of points in a plane, which don’t all lie on one line. Show that there exists one line, which goes through exactly two points.

The Solution (of L. M. Kelly) :

Lets assume, that there exists no such line. We pick a line L, that holds at least 3 points, so that there exists one point X which is not on the line, for which the distance between X and L is minimal. We denote the line, that goes through L and X and which is perpendicular to L with S. Since the line L holds at least 3 points, there exist at least 2 points R and Z on one side S.

Screen Shot 2014-12-22 at 2.20.16 PM

Now we assume, that R is the point further away form S. Then the distance between Z and the Line Through R and X is smaller, than the distance between X and L. This is a contradiction, which completes the proof.

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