Why would you use a geometric mean? It might make the problem go away, but why does it do that? what was the problem and why is a geometric mean the only solution?

I think it's a really bad strategy to respond to an apparent problem by just pulling ideas out of thin air until you find one who's flaws are non-obvious. It seems much more prudent to try to understand what the problem is and why it occurred, so that we can derive something that actually does the right thing. A problem is an indication that we don't understand what's going on, which should be a halt-and-rethink-everything moment.