Take AE midpoint f, AC midpoint g, and connect DF, FG and BG.

It is proved that FG is parallel to EC, and FG= 1/2EC= 1/2a.

Because BD = 1/2A and BD is parallel to EC, BD is parallel and equal to FG.

Since BD is perpendicular to BG(BD is perpendicular to face ABC), BDFG is rectangular and DF is parallel to BG.

It can be proved that BG is perpendicular to plane ACC 1, so DF is perpendicular to plane ACC 1, so plane ADE is perpendicular to plane ACC 1A 1.

(There should be other proofs for this. If there is anything simpler, please let me know. )

In (2), find the length of three sides.

AE= radical number 2 A is known by (1), DF is perpendicular to AE, and f is the midpoint of AE, so DE=DA= semi-radical number 5 A.

Then find the area (there are many methods), and find the root number of DF= half 6 a, and the square root of S = times the square of a.

As for the vector method, I won't go into details. A rectangular coordinate system can be established, with the midpoint H of BC as the coordinate origin, AH and BC as the X and Y axes, and the straight line parallel to BB 1 as the Z axis. You can do the rest yourself. ...