The crossratio porism Suppose that A,B,C,D are distinct collinear ppoints, 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 nonzero 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 hline.
(2)

