The cross-ratio porism Suppose that A,B,C,D are distinct collinear p-points, and that a,b,c,d are chosen so that A=[a], B = [b], C = [c] and D = [d]. Then we have (1) There exist unique non-zero real numbers α,β,γ and δ such that c = αa + βb, and d = γa + δb. (2) The ratio βγ/αδ does not depend on the choice of a,b,c,d.
proof As we remarked, part(1) is a simple consequence that {a,b} is a basis of the plane representing the common h-line. (2) Suppose that we choose other vectors a',b',c',d' with A = [a'], B = [b'], C = [c'] and D = [d']. Then there are non-zero numbers k,l,m,n such that a' = ka, b' = lb, c' = mc and d' = nd. Substituting from these into the equalities in (1), we get c' = (mα/k)a + (mβ/l)b, and d = (nγ/k)a + (nδ/l)b Using these, the required ratio is (mβ/l)(nγ/k)/(mα/k)(nδ/l) = βγ/αδ. Thus, the value obtained is independent of the choice of vectors.

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