First of all, in abc, the three-in-one advertisement is not only the angle bisector, but also the perpendicular bisector and height.

∫ Angle BAE= = Angle BDF? Angle abbe = = angle DBM.

∴△BEA∽△BMD

AB and BD are both in △ABD.

AB = ACBD = DC； AD⊥BC

And the angle ABC = 45 degrees.

∴? AD:BD:AB= 1:: root number 2

Then AE:MD=AB:BD= root number 2: 1.

So AE = root 2MD holds.