examday

/TMUA

A finite set of points in the plane is not contained in a straight line. The following proof shows that some straight line contains exactly two of the points.

Among all choices of a point $P$ from the set and a line $\ell$ through two other points, with $P$ not on $\ell$, choose one for which the perpendicular distance $PH$ is least. Here $H$ is the perpendicular foot on $\ell$.

Suppose that $\ell$ contains at least three points of the set.

[Missing argument]

This gives a smaller positive distance of the same kind, a contradiction. Hence $\ell$ contains exactly two points.

Which option correctly supplies the missing argument? In each option, $A,B$ must be points of the set on $\ell$.