On sets which meet each line in exactly two points

Using techniques from geometric measure theory and descriptive set theory, we prove a general result concerning sets in the plane which meet each straight line in exactly two points. As an application, we show that no such 'two-point' set can be expressed as the union of countably many rectifiable sets together with a set with Hausdorff 1-measure zero. Also, as a corollary, we show that no analytic set can be a two-point set provided that every purely unrectifiable set meets some line in at least three points. Some generalizations are given to 'n-point' sets and some other geometric constructions.