You can remove the line CD as it is not really helping, and instead introduce another line, parallel to CD, but passing through F.

On the face of it, there are too many degrees of freedom to be able to determine anything useful. HOWEVER, since we want DF to be a minimum, we can also add in the fact that angle EFD must be a right angle, and that lengths x and y must be equal.

Looking at the three similar triangles around the outside we can tell that:

64/x = y/z = w/v

As well as the facts that:

x = y; z + w = 64; 2x - v = 75

We can either do a bit of algebra and find a cubic in terms of x, which factorises to one real and two complex solutions.

Or we can try a few values for x and see what works. It turns out that x=y=48, z=36, w=28, v=21, and therefore DF = 35.