Nonuniqueness of Sixpartite Points

It is known that the area of any bounded, convex plane figure can be divided into equal sixths by three concurrent lines, and it is not hard to see that the same is true for perimeters. Calling the points of intersection of such lines area sixpartite points and perimeter sixpartite points, it is known that they are unique for triangles. We prove that they are not unique in general. Moreover, given any finite set of points in the plane we construct a convex polygon in which each of these points is an area sixpartite point, and a second polygon in which each is a perimeter sixpartite point.