Cross-Ratios on the Projective Line

Here are the coordinates of

Let
be any four points on this line. Their cross
ratio
is defined to be:

is a permissable value for the cross ratio. If the numerator and denominator both vanish at least three of the must be identical, so by l'Hôpital's rule the cross ratio is 1. The key property of the cross ratio is that it is invariant under collineations and changes of basis. In other words, it is a

See [23] for detailed proofs. In fact, invariance under collineations need only be verified on the projective line :

Show explicitly that the cross ratio is collineation invariant (Hint: ).