Before revision:  While we are still unable to solve this problem, in full generality, in subquadratic time, we obtain such a solution, in the algebraic decision-tree model, that uses only roughly O(n^{28/15}) constant-degree polynomial sign tests, for the special case where two of the sets lie on  one-dimensional curves and the third is placed arbitrarily in the plane.
After revision:  While we are still unable to solve this problem, in full generality, in subquadratic time, we obtain such a solution, in the algebraic decision-tree model, that uses only roughly O(n^{28/15}) constant-degree polynomial sign tests, for the special case where two of the sets lie on two respective one-dimensional curves and the third is placed arbitrarily in the plane.