Exercise 41, § 2

Let K be an ordered field, and E a vector space of dimension 2 over K. a) Show that the set of triples of pairwise distinct lines ($D_1, D_2, D_3$) passing through 0 has two orbits under the action of the group $\mathrm{GL}+(E)$ (consider the subgroup fixing two lines). b) The set of triples of pairwise distinct half-lines ($A_1, \Delta_1, A_1$) with origin 0 has 7 orbits under the action of $\mathbf{GL}(E)$, and 14 under the action of $\mathrm{GL}+(E)$ (same method).