Let $\mathcal{A}$ be an affine plane with a finite amount of points on each line. Suppose that Desargues' theorem holds in $\mathcal{A}$. Then it is known that we can associate a division ring $\mathbb{K}$ to $\mathcal{A}$ such that $\mathcal{A}$ can be (with a point $o$ as center) considered as a $2$-dimensional $\mathbb{K}$-vector space.
Because $\mathbb{K}$ is finite, by Wedderburns little theorem, $\mathbb{K}$ is a field. So Pappus's hexagon theorem holds automatically. My question is whether we can directly prove Pappus's theorem, only making use of the fact that $\mathcal{A}$ has a finite amount of points on each line.
Tell me if this terminology or approach is non-familiar. I appreciate every answer or suggestion.