Let S be a set in a two dimensional Euclidean space E2. Such a set S is said to be arcwise convex [5] if each pair of its points can be joined by a convex arc lying in S. A convex arc is, by definition, an arc which is contained in the boundary of a plane convex set. In two previous papers, [5; 6 ], the author studied certain properties of closed arcwise convex sets. It is the purpose of this paper to study an interesting class of sets which satisfy the three point arcwise convexity property, defined as follows: DEFINITION 1. A set SCE2 is said to have the three point arcwise convexity property if each triple of points x CS, y CS, zCS is contained in a convex arc belonging to S. It should be observed that the above property implies there exists a convex arc in S having two of the three points x, y, z as end points and the third point in its interior. DEFINITION 2. A convex curve, as distinguished from a convex arc, is a closed connected portion of the boundary of a plane convex set. A convex curve may have two, one, or no end points, and it may be bounded or unbounded. The following theorem characterizes the closed sets in E2 which have the three point arcwise convexity property.