Discrete Helly type theorems

We show the following discrete Helly-type results in the plane. Let P be a set of points and D a family of convex pseudodisks in the plane s.t. every triple of pseudodisks in D intersects at a point in P. Then, two of the points in P hit all pseudodisks in D. Three similar results that we prove with the roles of points and regions exchanged are the following: let P be a set of points and H a set of halfspaces in the plane s.t. every triple of points from P is covered by some halfspace in H. Then, two of the halfspaces in H cover all points in P. If, instead, every pair of points in P is covered by some halfspace in H, we show that three of halfspaces in H suffice to cover all points in P. Finally, we show that if every pair of points is covered by an axis-parallel rectangle, then five rectangles are required to cover all points.