Perfect Hexagons, Elementary Triangles, and the Center of a Cubic Curve

If six points in the plane are labeled with Z(6) so that for each k in Z(6) the set of lines W-k = {(a, b) : a+b = k} concurs at a point X-k then the six points form a perfect hexagon P. The vertices of P and the perspective points {X-k : k is an element of Z(6)} lie on a cubic curve. If we complete P by including all lines which join vertices of P as well as all intersection points of these lines, we obtain a figure which contains many perfect hexagons. We develop a theory of cubic curves which explains this phenomenon.