There are not too many magic configurations (Q2482202)

A finite planar point set \(P\) is said to be a magic configuration if there is a correspondence from \(P\) to a set of positive numbers (called weights) such that, for every line \(l\) determined by \(P\), the sum of the weights of all points of \(P\) on \(l\) equals \(1\). Consider a figure constituted by the three vertices of a regular triangle, its centre, and the three midpoints of the sides. Assigning \(\frac{1}{4}\) to the vertices and the centre, and assigning \(\frac{1}{2}\) to the midpoints, one obtains a magic configuration, called failed Fano configuration. The authors prove a conjecture of Murty from 1971 saying that apart from failed Fano configurations, every set of \(n\) points that is a magic configuration is either in general position, or contains \(n-1\) collinear points.