The angle ANB is equal to 90 and AN=BN.
And because AM=BM=4 root number 2.
So the radical number 2 of AB=8
So AB=BN=8 (Pythagorean theorem)
And because angle A= angle b = 45 and angle CMD=45.
So AME angle plus BMF angle = 135.
Because angle AME plus angle AEM = 135.
So angle AEM= angle BMF (equivalent replacement)
So the triangle AEM is similar to the triangle BMF.
So AE: BM = am: BF
That is, 3: 4 root number 2=4 root number 2: BF
BF=32/3
So FN=BN-BF
EN=AN-AE
Then Pythagorean theorem, find EF.
The square of EF = the square of en plus the square of FN
Final prescription
That's the process. As for the result, I have no time to forget it. sorry O(∩_∩)O~