The ABC theorem

Theorem Let $A,B,C$ be subgroups of $G$. Let $B \le A \le G$ and $C \le G$, then we have:
A) $A \cap BC = B(A \cup C)$ as sets.
B) if $B \unlhd A$ then $B \cap C \unlhd A \cap C$ and $(A \cap C)/(B\cap C) \simeq B(A\cap C)/B$
C) If $B \unlhd A$ and $C \unlhd G$ then $BC \unlhd AC$ and $AC/BC \simeq A/B(A\cap C)$.
proof hint: Use Noethers isomorphism theorems and sometimes consider elements.