Theorem PI1 If A,B,C,D,E are collinear ppoints such that (A,B,C,D) = (A,B,C,E), then D = E.


proof Observe that, as the crossratios are defined, we must have A,B,C,D,E collinear. Suppose that A = [a], B =[b], C = [c], D = [d] and E = [e]. As before, we have real constants such that c = αa + βb, d = γa + δb and d = λa + μb. Then, by the definition of cross ratio, (A,B,C,D) = βγ/αδ and (A,B,C,E) = βλ/αμ. As the crossratios are equal, βγ/αδ = βλ/αμ, so λ = γμ/δ. Then e = (γ&mu/δ)a + μb = (μ/δ)(γa + δb) = (μ/δ)d. Since e is a multiple of d, E = D, as required.

