A generalised Sylvester-Gallai Theorem

Abstract

We give an algorithmic proof for the contrapositive of the following theorem that has recently been proved by the authors:

Let S be a finite set of points in the plane, with each point coloured red, blue or with both colours. Suppose that for any two distinct points A and B in S sharing a colour k, there is a third point in S which has (inter alia) the colour different from k and is collinear with A and B. Then all the points in S are collinear.
This theorem is a generalization of both the Sylvester-Gallai Theorem and the Motzkin-Rabin Theorem.