Proof: Because △ABC is a right triangle.

So the angle BAC=90 degrees.

After connecting AP, AP=CP=BP (because P is the midpoint of BC).

AC=AB=2, angle ACB= angle ABC=45 degrees.

So the angle APC=90 degrees

Because the angle EPF=90 degrees

So angle CPN= angle APM

Chamfer again to get angle PAB=45 degrees.

So angle PAM= angle PCN

In △CPN and △APM,

Because: angle CPN= angle APM

AP=CP

So angle PAM= angle PCN

So △ CPN △ APM

So PN=PM