I am looking for an elementary proof of the following lemma, which concerns what Green and Tao call "triangular grids" (see arXiv:1208.4714).
Let $a_1$, $a_2$, $a_3$, $a_4$, $b_1$, $b_2$ be six arbitrary lines in the plane (in general position, for whatever the appropriate sense of "general position" is).
Let $c_1$ be the line through the intersection points $a_3b_1$ and $a_2b_2$.
Let $c_2$ be the line through $a_4b_1$ and $a_3b_2$.
Let $b_3$ be the line through $a_1c_1$ and $a_2c_2$.
Let $c_3$ be the line through $a_4b_2$ and $a_3b_3$.
Let $b_4$ be the line through $a_1c_2$ and $a_2c_3$.
Let $c_4$ be the line through $a_4b_3$ and $a_3b_4$.
Let $b_5$ be the line through $a_1c_3$ and $a_2c_4$.
Let $c_5$ be the line through $a_4b_4$ and $a_3b_5$.
Then the intersection points $b_1c_3$, $b_2c_4$, $b_3c_5$ are collinear.
As a consequence, this triangular grid can be indefinitely continued.
See figure below.
This can be proven using cubic curves, specifically the Cayley-Bacharach theorem: In the dual setting, all the points dual to the given lines lie on a common cubic curve. See the above-mentioned paper of Green and Tao.
But as I said, I was wondering whether an elementary proof exists.
I see there is previous literature on these triangular grids (also called with other names; see reference  in the above paper, and references cited there). I quickly glanced at the literature, and I didn't find what I'm looking for.