Detailed steps:

(1), because △ADC and △BCE are isosceles right triangles.

So so, so so.

So am/AE = DM/MC = AC/AB, CN/CE = DN/DB = AC/AB,

So DM/DC=DN/DB, which is MN//BC.

(2) △CMN from (1) is an isosceles right triangle.

So MN=√2MC

And because MC/DC=BC/AB.

MC=BC/(AC+BC)×DC

=BC/(AC+BC)×√2/2AC

So 1/MN=(AC+BC)/BC×AC.

This is the evidence.

(3)AC/MN=AE/ME=AB/BC

BC/MN=BD/DN=AB/AC，

So the two formulas are added together,

AB/MN = AB/BC+AB/AC = 1+AC/BC+ 1+BC/AC≥4

So MN≤ 1/4AB