(1), as shown in the figure, m is the midpoint of AC, and triangle ABC and ADC are right triangles.

So BM=DM

N is the midpoint of BD.

So MN is perpendicular to BD (the center line of isosceles triangle, the bisector of the top angle and the bottom height are one).

(2) Because the angle BAC=30? Angle CAD = 45

Triangle ABM and ADM are isosceles triangles.

So angle MBD+ angle MBM= 180? -2*(30? +45? )=30?

And BM=DM, then the angle MBD= 15?

And because BM= 1/2AC=2 (the median line on the hypotenuse of a right triangle is equal to half of the hypotenuse).

So MN=BM*sin 15? = (the root of 6 is the square root of 2) /2