I would start by expanding all three sets of brackets. See what you can do from there...
Let's dissect the question a little bit.
If we expand those brackets, we find:
a^2 + ac + ab + bc = b
b^2 + ab +bc +ac = c
c^2 + bc + ac + ab = a
Note that all 3 equations have the same 3 terms : ab, ac and bc. So let's say "Let x = ab + ac + bc". Then our equations look more like:
a^2 + x = b
b^2 + x = c
c^2 + x = a
Does it become more clear now how to show this?