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COURSE OF PROJECTIVE GEOMETRY

§ 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 *D*_{2}D_{3} on *A*_{2}A_{3} is the fourth harmonic with {*A*_{2}, A_{3}} and *D*_{1}.
Likewise the intersection point of *E*_{2}E_{3} and *A*_{2}A_{3}.

*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*.

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