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

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