There are I think at least two views on the cross ratio of 4 points even https://en.wikipedia.org/wiki/Cross-ratio gives two definitions
And I was wondering what is the relation between the two
The Geometrical view:
The cross ratio of 4 points A,B;X,Y is
$$ CR_g =\frac{d(A,Y) d(B,X)}{d(A,X) d(B,Y)}$$
Where d(I,J) is the distance between I and J.
The Algebraic view:
The cross ratio of four (complex) numbers a,b,x,y is:
$$ CR_a=\frac{(y-a) (x-b)}{(x-a)(y-b)}$$
Wondering / Questions
Points in a plane can be represented by complex numbers and this made me wonder:
What is the relationship between the two cross ratio views if $A,B,X,Y$ and $a,b,x,y $ represent the same points.
And I also would like to see a proof of that relation.
Explorations:
When the points A,B,X,Y are on a line then the cross ratio's are equal
When the points are on a circle $CR_a$ is a real number (see reference in comments)