Let O_{1}, O_{2}, \ldots, O_{2012} be 2012 circles in the plane such that no circle intersects or contains any other circle and no two circles have the same radius. For each 1 \leq i<j \leq 2012, let P_{i, j} denote the point of intersection of the two external tangent lines to O_{i} and O_{j}, and let T be the set of all P_{i, j} (so |T|=\binom{2012}{2}=2023066). Suppose there exists a subset S \subset T with |S|=2021056 such that all points in S lie on the same line. Prove that all points in T lie on the same line.