 Eight-point cubic completion for two triples of lines

Let two triples of projective lines meet in nine distinct points. Any plane cubic containing eight contains all nine. To prove this, write the triples as $F=L_1{L}_2{L}_3$ and $G=M_1{M}_2{M}_3$, with the missing point on $L_3$. The candidate cubic $H$ agrees with a scalar multiple of $G$ on $L_1$, so $H-\lambda G=L_{1}Q$. Its three known zeros on $L_2$ force $Q=L_{2}L$. Two remaining known zeros on $L_3$ force the linear factor $L$ to be a multiple of $L_3$. Hence $H=\lambda G+\mu F$ and vanishes at the missing point.