I used the letter 'u' instead of 'x' for CE because I used an xy coordinate system making ABCD a unit square with A at the origin.  Figured the equation for the line which includes segment DE.  Figured the equation of a perpendicular line through the origin and the intersection point.  From there determined the distance from the origin to the line segment:  this was sqrt(1+u^2) / (1+u^2).

And figured the length of DE:  sqrt(1+u^2).

So, the area of the square was 1, and

the area of the rectangle was sqrt(1+u^2)* sqrt(1+u^2) / (1+u^2).

Also 1.

So the ratio is 1 and independent of the length of CE.

Details:

A = (0,0)

B = (0,1)

C = (1,1)

D = (1,0)

E = (1-u, 1)

line DE:  y = -(1/u)x + 1/u

line perpendicular to DE and through origin:  y = ux

point of intersection:  (1/(1+u^2), u/(1+u^2)) from which

distance from A to line DE: sqrt(1+u^2) / (1+u^2)