= Eight-point cubic completion for two triples of lines
{title2=$H=\lambda G+\mu F$}
= Chasles nine-point theorem
{c}
{synonym}
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 $F=L_1L_2L_3$ and $G=M_1M_2M_3$, with the missing point on $L_3$. The candidate cubic $H$ agrees with a scalar multiple of $G$ on $L_1$, so $H-\lambda G=L_1Q$. Its three known zeros on $L_2$ force $Q=L_2L$. 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.
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