proof of the cross-ratio theorem

The projective cross-ratio theorem
Cross-ratio is invariant under the projective group P(2).

Suppose that A = [a], B = [b], C = [c] and D = [d] are distinct, collinear p-points,
and that t is a projective transformation.

As t is projective, A' = t(A), B' = t(B), C' = t(C) and D' = t(D) are distinct and collinear.

Now, t has the form t([x]) = [Mx], where M is a non-singular 3x3 matrix, so we have
A' = [Ma], B' = [Mb], C' = [Mc] and D' = [Md].

As in the porism, c = αa + βb, and d = γa + δb.
The map x -> Mx is linear, so that Mc = αMa + βMb, and Md = γMa + δMb
Hence the ratio for (A',B',C',D') is βγ/αδ, i.e. is equal to that for (A,B,C,D).

projective cross-ratio page