lemma 3
If a line cuts the lines of a pencil with vertex P at the points A,B,C,D,
then (PA,PB,PC,PD) = sin(<APC)sin(<BPD)/sin(<BPC)sin(<APD).

proof
We know that the cross-ratio is that defined by the points A,B,C,D.
The proof of lemma 1 indicates that this depends on numbers such that
cab, dab where a,b,c,d are the position vectors of A,B,C,D.
It is gven by
(*)          {axc/bxc}/{axd/bxd}.

As A,B,C,D are collinear, α+β=γ+δ=1. Now cab, dab, so
c-p=α(a-p)+β(b-p), d-p=γ(a-p)+δ(b-p).
Thus we can replace a,b,c,d in (*) by the vectors PA,PB,PC,PD

We usually define uxv by its magnitude |u||v|sinθ, where θ is the angle
between u and v, measured in the range [0,π], and direction perpendicular
to u,v, with {u,v,uxv} right-handed.
If we suppose that u and v lie in the xy-plane, then uxv is parallel to z, the
unit vector in the z-direction. Then uxv is (|u||v|sinφ)z, where φ is the angle
between u and v measured in the anticlockwise direction.

When we use the vectors PA,PB,PC,PD, the lengths cancel, giving the result.