A Helly-type theorem for simple polygons

Let P be a family of simple polygons in the plane. If every three (not necessarily distinct) members of P have a simply connected union and every two members of P have a nonempty intersection, then boolean AND{P:P in P} not equal 0. Applying the result to a finite family C of orthogonally convex polygons, the set boolean AND{C:C in C} will be another orthogonally convex polygon, and, in certain circumstances, the dimension of this intersection can be determined.