Solution
= Solution
Let $u=(1,1,1)$ and $v=(1,\alpha,\beta)$. The vector representing $r_{23}$ satisfies
$$
(\alpha\beta-1,\alpha(\beta-1),\beta(\alpha-1))=\alpha\beta\,u-v.
$$
Consequently the three representative vectors are <linearly dependent>, so \b[$r_{12},r_{13},r_{23}$ are collinear] in the <projective plane>. Equivalently their three-by-three <determinant> vanishes. This is the two-line configuration of <Pappus's hexagon theorem>.