BG=AE=x，

Because EF=BF=y, so

ef^2=eg^2+gf^2= 12^2+(y-x)^2=y^2

simplify

y=x/2+72/x

(2) because EF=BF, so ∠FEB=∠FBE=∠AEB,

Then, fold △ABE along the line where BE is located, and point A falls on EF, that is, A' is on EF and BA' is perpendicular to EF.

If △A'BF is an isosceles triangle, BF=√2BA'

that is

y=x/2+72/x= 12√2

Solve? x= 12(√2- 1)

So AE= 12(√2- 1)