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
As in the porism,
c = αa + βb, and d = γa + δb.
projective cross-ratio page