Reduction from PARTITION to 3PARTITION
I guess that this question should not be left floating around unanswered.

The reasoning in (b) can't be used to prove that this problem is NP-complete because, at best, it implies the reduction from 3PARTITION to PARTITION which is of no use. Any problem in NP can be reduced to PARTITION. 

The proper way of doing this (after showing that the problem is in NP) is to: 
i) Start from an arbitrary instance of PARTITION 

$A=w_1,...,w_n$ with $S(A)=\sum\limits^{n}_{i=1}w_{i}=2m$

ii) Append $m$ to $A$ and obtain $A'$, an instance of the 3PARTITION problem.
iii) Prove that $A\in PARTITION\Longleftrightarrow A'\in 3PARTITION$
iv) Conclude that $PARTITION \leq_{p}3PARTITION$