Typing is quite troublesome. Let me say it briefly. Mainly to get the angle relationship.

Angle mna=45, and half of angle A's mna is 15, so angle AMN is 120, so DMB is 60 degrees.

That is to say, Triangle Yade and Triangle BDM are similar triangles with three angles of 90 degrees, 30 degrees and 60 degrees respectively.

Then I know that BD=4, so I can find it by trigonometric function.