Suppose f is the cubic polynomial vanishing on the three lines through AB, CD, EF and g is the cubic vanishing on the other three lines BC, DE, FA. Pick a generic point P on the conic and choose λ so that the cubic h = f + λg vanishes on P. Then h = 0 is a cubic that has 7 points A, B, C, D, E, F, P in common with the conic. But by Bézout's theorem a cubic and a conic have at most 3 × 2 = 6 points in common, unless they have a common component.