Let's say I have a particular finite presentation and want to show it's actually a presentation for the group I claim it's a presentation for. That group might be specified, say, by a linear or permutation representation, or more generally as the automorphism group of some object. (Obviously the problem of showing two different presentations are equivalent is well-studied already and I know where to look for that.)
Of course there's no general technique here that's of any use. I'm just interested in seeing nice proofs in particular cases, to get an idea of the different ways it can be done.