It is proved that by connecting MC, it is easy to know that the quadrilateral MECF is a moment, so EF=MC.

Because point A and point C are symmetrical about DB, AM=MC.

That is AM=MC=EF.

It can also prove that

AB=BC, & ltABM = & ltMBE=45 degrees, MB=BM.

Triangle ABM and triangle MBE congruence (SAS)

So AM=MC

Because EF=MC

That is AM=EF.