Past exam of the short options course of the University of Oxford / 2021 / 3 / c / ii / Solution

Choose a point $A$ off $L$, then a point $B$ on $A{R}_2$ distinct from $A,R_2$. PutThese four points form a projective frame: $A,B$ lie off $L$, and $AB=A{R}_2$ meets $L$ at neither $R_0$ nor $R_1$. Form, using only joins and intersections,Here the point called $Q_1$ in (i) is $R_2$, while $Q_3$ is $R_3$. Thus (i) gives $(R_0{R}_1:R_3{R}_2)=-1$. Interchanging the last two arguments reciprocates the cross-ratio, soAll auxiliary choices are possible over a field of characteristic zero, which is infinite. Uniqueness from (a)(ii) shows that the final point is independent of those choices. The picture uses an affine chart in which every relevant intersection is finite; parallel-line exceptions in another chart are ordinary intersections on the projective line at infinity.