Because the side length of the square ABCD is 4

So AB=BC=4

Angle ABC=90 degrees

AB parallel DM

So angle EBF= angle ECM

Angle BFE= angle m

Because e is the midpoint of BC

So BE=CE= 1/2BC=2.

So triangle BFE and triangle CME are congruent (AAS)

So BF=CM

EF=EM

Because FEG angle+Meg angle = 180 degrees.

Angle FEG=90 degrees

So FEG angle = Meg angle =90 degrees.

Because EG=EG

So triangle FEG and triangle Meg are congruent (SAS)

So GF=GM

Angle HFE= angle m

So Angel ·BFE = Angel ·HFE

Because CH is parallel to EF

So GF/FH=GM/CM

So FH=CM

So BF=GF

Because EF=EF

So triangle BFE and triangle HFE are congruent (SAS)

So angle ABC= angle EHF=90 degrees.

Because AH is vertical FG

So the angle AHF=90 degrees

So angle AHF+ angle EHF= 180 degrees.

So the three-point line of a, h, e and e.

So the triangle ABE is a right triangle.

So AE 2 = AB 2+BE 2.

So AE=2 times the root number 5.

Because angle ABC= angle AHF=90 degrees

Angle BAE= angle BAE

So the triangle BAE is similar to the triangle HAF (AA).

So AF/AE=GF/BE

Because AF=AB-BF

So (4-BF)/2 times the root number 5=BF/2.

So BF= radical number 5- 1.