Problem Statement
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?
Categories:
Geometry
Progress
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].
Source: erdosproblems.com/735 | Last verified: January 16, 2026