SIMON-MARAIS 2024 B3

Let $\mathcal{L}$ be the set of all lines in the plane and let $\mathcal{P}$ be the set of all points in the plane. Determine whether there exists a function $g : \mathcal{L} \rightarrow \mathcal{P}$ such that for any two distinct non-parallel lines $\ell_1, \ell_2 \in \mathcal{L}$, we have

(a) $g(\ell_1) \neq g(\ell_2)$, and

(b) if $\ell_3$ is the line passing through $g(\ell_1)$ and $g(\ell_2)$, then $g(\ell_3)$ is the intersection of $\ell_1$ and $\ell_2$.