Solution:

Let $AD = h,$ and place $A = (0, 0),$ $B = (100, 0),$ $C = (100, h),$ $D = (0, h),$ so $E = \left(0, \frac{h}{2}\right).$ Line $AC$ has slope $\frac{h}{100}$ and line $BE$ has slope $\frac{h/2}{-100} = -\frac{h}{200}.$ Perpendicularity gives $$\frac{h}{100} \cdot \left(-\frac{h}{200}\right) = -1,$$ so $h^2 = 20000$ and $h = 100\sqrt{2} \approx 141.42.$

The greatest integer less than $AD$ is $141.$