In the cube abcd-a.b.c.d., e is the midpoint of AA. Proof: A.C flat plane BDE

In the cube abcd-a.b.c.d., e is the midpoint of AA. Proof: A.C flat plane BDE

Make auxiliary line AC to BD to f and connect EF
∵ e is the midpoint of AA and F is the midpoint of AC
The EF is the median line of the triangle ACA
A. C is parallel to ef
∵ EF in plane BDE
The A.C is parallel to the plane BDE

In the square prism abcd-a1b1c1d1, Aa1 = 2Ab, e is the midpoint of CC1

(1) It is proved that: connect AC, let AC ∩ BD = O. from the condition, ABCD is a square, so o is the midpoint of AC. ∩ e is the midpoint of CC1, ∩ OE ⊂ plane BDE, AC1 ⊈ plane BDE. ∩ AC1 ⊂ plane BDE. (2) connect b1e. Let AB = a, then in △ bb1e, be = b1e = 2A, BB1 = 2A. ∩ be2 + B