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Coxeter- Figure 4.2B

ABCX is perspective onto A1B1C1X1 through point O, which in turn is perspective onto GNMQ through point A', which is in turn perspective onto A'B'C'X' through point A1. Therefore, seven points on a line, A, B, C, X, A', B', and C' determine a unique eighth point, X', because otherwise, it would lead to a contradiction. This leads to the Fundamental Theorem Projective Geometry : That a projectivity is determined when three collinear points and the corresponding three collinear points are given.