proof of the cross-ratio porism

The cross-ratio porism
Suppose that A,B,C,D are distinct collinear p-points, and that a,b,c,d are chosen
so that A=[a], B = [b], C = [c] and D = [d]. Then we have
(1) There exist unique non-zero real numbers α,β,γ and δ such that
c = αa + βb, and d = γa + δb.
(2) The ratio βγ/αδ does not depend on the choice of a,b,c,d.

As we remarked, part(1) is a simple consequence that {a,b} is a basis of the plane
representing the common h-line.

Suppose that we choose other vectors a',b',c',d' with
A = [a'], B = [b'], C = [c'] and D = [d'].
Then there are non-zero numbers k,l,m,n such that
a' = ka, b' = lb, c' = mc and d' = nd.
Substituting from these into the equalities in (1), we get
c' = (mα/k)a + (mβ/l)b, and d = (nγ/k)a + (nδ/l)b
Using these, the required ratio is (mβ/l)(nγ/k)/(mα/k)(nδ/l) = βγ/αδ.
Thus, the value obtained is independent of the choice of vectors.

projective cross-ratio page