(2) If BM is parallel to plane AEF and plane BGM is parallel to plane AEF, then GM is parallel to AE; (Two parallel planes are parallel to the straight line cut by the third plane)

(3) From the above, it can be seen that GM is parallel to AE and G is the midpoint of EC, so GM is a midline of triangle AEC, so M is the midpoint of AC. (midline theorem)

The method is as above. For specific theorems, you can find the original text in the math book.