If z1, z2, z3, z4 are complex numbers, then the quantity:

(z3-z1)   (z4-z2)  
------- * -------
(z3-z2)   (z4-z1)

is called the cross ratio of the 4-tuple (z1,z2,z3,z4).
The cross ratio is conserved under translation (z+a), inversion (1/z), dilation and rotation (a*z). For such a transformation s, the cross ratio of (s(z1),s(z2),s(z3),s(z4)).

In the complex numbers, any transformation s that is a composition of these four simple transformations is of the form:

       a*z + b
s(z) = -------
       c*z + d
where a,b,c,d,z are complex numbers.

From this it is simple calculation to verify that the cross ratio is conserved.