Desargues configuration

Example

The Desargues configuration is a configuration $(10_3)$ in the projective plane. Take two triangles $a_1 a_2 a_3$ and $b_1 b_2 b_3$ in perspective from a point $O$, so that $O$, $a_i$, $b_i$ are collinear for $i = 1, 2, 3$. Let $c_{ij}$ be the intersection of the sides $a_i a_j$ and $b_i b_j$. By Desargues's theorem, the three points $c_{12}$, $c_{13}$, $c_{23}$ lie on a line, the axis.

Its $10$ points are $O$, the six vertices and the three points $c_{ij}$. Its $10$ lines are the three lines $O a_i b_i$, the six sides of the triangles and the axis. Check: each line has $3$ of the points, for instance the side $a_1 a_2$ contains $a_1$, $a_2$, $c_{12}$. Each point lies on $3$ lines. For instance, $O$ lies on the lines $O a_i b_i$, the vertex $a_1$ on $O a_1 b_1$, $a_1 a_2$, $a_1 a_3$, and $c_{12}$ on $a_1 a_2$, $b_1 b_2$ and the axis.

A combinatorial model: the points are the $2$-element subsets of $\set{1, 2, 3, 4, 5}$ and the lines are the $3$-element subsets, with incidence by inclusion. Each $3$-set contains three $2$-sets, and each $2$-set lies in three $3$-sets. Its incidence graph is the Desargues graph.