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

Abstract

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 perfecthexagonP. The vertices of P and the perspectivepoints {X _k : k ∈ 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.