So triangle BEC and triangle BAFF are congruent.
So FA=EC
BF=BE
Angle ABF= angle CBE
Because the angle ABC=90 degrees
AB=AC
So the triangle ABC is an isosceles right triangle.
So angle BAD= angle BCE=45 degrees
So the angle BAF=45 degrees
Because angle ABC= angle ABD+ angle DBE+ angle CBE=90 degrees.
Angle DBE=45 degrees
So angle ABF+ angle ABD= angle DBF=45 degrees.
So angle DBF= angle DBE=45 degrees.
Because BD=BD
So triangle DBF and triangle DBE are congruent (SAS)
So DF=DE
Because angle DAF= angle BAF+ angle BAD=45+45=90 degrees.
So the triangle DAF is a right triangle.
From Pythagorean Theorem:
DF^2=AF^2+AD^2
So de 2 = ad 2+EC 2.
(3)MN=AM+CN
Proof: extend DA to make AF=CN and connect BF.
Because the quadrilateral ABCD is a square
So angle BAF= angle BCN=90 degrees.
Angle ABC= Angle ABM+ Angle MBN+ Angle CBN=90 degrees.
AB=BC
So triangular Baff and triangular BCN are congruent (SAS)
So BF=BN
Angle ABF= angle CBN
So angle ABF+ angle ABM+ angle MBN=90 degrees.
Because the angle MBN=45 degrees
So ABF angle ABM angle = MBF angle =45 degrees.
So angle MBF= angle MBN=45 degrees.
Because BM=BM
So triangle BMF and triangle BMN are congruent (SAS)
So MF=MN
Because MF=AM+AF
So MN=AM+CN