SOLVED

Given any $n$ points in $\mathbb{R}^2$ when can one give positive weights to the points such that the sum of the weights of the points along every line containing at least two points is the same?

A problem of Murty, who conjectured this is only possible in one of four cases: all points on a line, no three points on a line, $n-1$ on a line, and a triangle, the angle bisectors, and the incentre (or a projective equivalence).

The previous configurations are the only examples, as proved by Ackerman, Buchin, Knauer, Pinchasi, and Rote [ABKPR08].