The first step is to distribute the 6abc into the three terms on the left side. Consider that case where a>b>c. Create a new variable for a-b and a-c. Then you can simplify the proof using your initial assumption. Since the proof is symmetric in abc, solving for a>b>c is sufficient. Well you also need to consider equities, but that only makes the proof easier.
Let me know if that helps or you need a bit more guidance.