In Search of Melchior’s Ordinary Points

In 1893, James Joseph Sylvester posed the following problem: given n not all collinear points in the plane, must there be a line determined by two of the points that does not pass through any of the other points? In 1941, Eberhard Melchior studied the equivalent dual problem in the projective plane: given a set of n lines in the (real) projective plane, not all passing through a common point, must there be a point where exactly two of the lines intersect? Such a point of intersection is called an ordinary point. Via a clever double counting combinatorial argument, Melchior found that, in fact, there must be three such points. Given the many simple “visual” proofs of what is today known as the Sylvester-Gallai Theorem—the theorem that states there must be at least one ordinary point—a natural question is whether there is a simple visual proof that recovers all three of Melchior’s ordinary points. This paper provides such a proof.