So angle DMF= angle DME=90 degrees.
Angle DNE= Angle DNC=90 degrees
Because the trapezoid is folded along de.
So CD=FD
EF=EC
Angle definition = angle deviation
Because angle B=90 degrees
So angle B= angle DNC=90 degrees.
So AB parallel DN
Because AD and BC are parallel
So the quadrilateral ABND is a parallelogram.
Because AD=AB=2
So the quadrilateral ABND is a diamond.
So the quadrilateral ABND is a square.
So the angle A=90 degrees.
AD=DN=AB=BN
Because BC=BN+CN=3.
So CN=3-2= 1.
In the right triangle DNC, the angle DNC=90 degrees.
From Pythagorean Theorem:
CD^2=DN^2+CN^2
So FD=CD= root number 5
In the right triangle ADF, the angle α= 90 degrees.
From Pythagorean Theorem:
FD^2=AD^2+AF^2
So AF= 1
Because AB=AF+BF=2
So AF=BF= 1.
So f is the midpoint of AB.
So (1) this conclusion is correct.
(2) Proof: Because angle DME= angle DNE=90 degrees (proved)
Angle Definition = Angle Deviation (Confirmed)
DE=DE
So triangle DME and triangle DNE are congruent (AAS)
So DM=DN
Because AD=DN
So AD=DM
Because angle α = angle DMF=90 degrees
So triangle ADF and triangle MDF are right triangles.
Because DF=DF
So right triangle ADF and right triangle MDF are congruent (HL)
So angle AFD= angle EFD
So FD bisects the AFE angle.
So (2) this conclusion is correct.
(3) Solution: Because the S triangle ADF= 1/2AD*AF.
AD=2 AF= 1
So the s triangle ADF= 1.
Because the s triangle BEF= 1/2*BE*BF.
BE=4/3
BF= 1
So the S triangle BEF=2/3.
So s triangle ADF+S triangle BEF=5/3.
Because angle B=90 degrees
So the BEF triangle is a right triangle.
So from the Pythagorean theorem,
EF^2=BE^2+BF^2
So EF=5/3.
Because the S triangle DEF= 1/2EF*DM.
DM=AD=2
So s triangle DEF=5/3.
So s triangle DEF=S triangle ADF+S triangle BEF
So (3) this conclusion is correct.
(4) solution: because CE=BC-BE
BC=3
BE=4/3
So CE=5/3
Because NE=BN-BE
BN=AD=2
BE=4/3
So NE=2/3
In the right triangle DNE, the angle DNE=90 degrees.
So from the Pythagorean theorem:
DE^2=DN^2+NE^2
So DE=2 times the root number 5/3.
Because CD= root number 5
CE=5/3
So de, CD and be are not equal.
So the conclusion that the triangle DEC is an isosceles triangle is wrong.
To sum up, there are three correct conclusions.
So choose C.