The projective crossratio theorem Crossratio is invariant under the projective group P(2).


proof Suppose that A = [a], B = [b], C = [c] and D = [d] are distinct, collinear ppoints, 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 nonsingular 3x3 matrix, so we have
As in the porism,
c = αa + βb, and d = γa + δb.

