§ 12:

O48
Let λ = -1. Then, according to O36, we also get -1 with
1 2 3 4 --- 2 1 4 3 --- 3 4 1 2 --- 4 3 1 2
1 2 4 3 --- 2 1 3 4 --- 3 4 2 1 --- 4 3 2 1 .
This is important for being able to define the concept of pairs {1,2} and {3,4} that separate harmonically.

O49
Start with a point D outside l, and then draw lines as below. The numbers indicate the order. There exist other possible constructions.

O50

The intersection point D₂D₃ on A₂A₃ is the fourth harmonic with {A₂, A₃} and D₁. Likewise the intersection point of E₂E₃ and A₂A₃.

O51
G := AP.BC. If {P,F} and {D,E} separate harmonically, then so do {B,G} and {C,D}.
So construct D with B, C, G and O49.