Intersection point of lines in projective geometry

Question

Let $K$ be a field and let $g_1,g_2,g_3,h_1,h_2,h_3,$ be different lines in the projective plane $\mathbb{P}^2(K)$, so that $g_1,g_2,g_3$ have one intersection point $A$ and $h_1,h_2,h_3$ have another intersection point $B$. $A \notin h_1,h_2,h_3$ and $B \notin g_1,g_2,g_3$. For $i,j \in \{ 1, 2, 3 \}$ let $C_{ij}$ be the intersection point in $g_i \cap h_j$.

Do the lines $(C_{12}C_{21}), (C_{13}C_{31}), (C_{23}C_{32})$ have an intersection point?

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I have thought that maybe $A,B,C_{11} $ are not collinear but I could't find an answer using this, help (with spoilers) is appreciated!

Answer 1

The question is equivalent to its dual:

Let $g_1,g_2,g_3$ be three distinct points on a line $A$, and let $h_1,h_2,h_3$ be three distinct points on a line $B$ and suppose $A\cap B\notin\{g_1,g_2,g_3,h_1,h_2,h_3\}$.

Let $C_{ij}$ be the line $(g_i,h_j)$. Are $C_{12}\cap C_{21}$, $C_{13}\cap C_{31}, C_{23}\cap C_{32}$ collinear?

But this is just Pappus' theorem. If you want to visualise it, you can assume that $A\cap B$ is the point at infinity. Or, equivalently, that $A\parallel B$ in the affine plane $\Bbb A^2(K)$.