Radial Points in the Plane

A radial point for a finite set P in the plane is a pointq ≠ ∈ P with the property that each line connecting q to a point ofP passes through at least one other element of P. We prove a conjecture of Pinchasi, by showing that the number of radial points for a non-collinear n -element set P is O(n). We also present several extensions of this result, generalizing theorems of Beck, Szemerédi and Trotter, and Elekes on the structure of incidences between points and lines.