Great information here. I just wanted to add one exercise that I feel does a great job strengthening all of the supporting muscles in my legs – slacklining. I think any balance exercise would work just as well, but I had the strongest legs of my life after the few months that I was really into slacklining. With recent backpacking knee pain I've started doing it again – so far, so good. But I haven't been down the Grand Canyon in a couple of months, so I haven't had a real test.
The downside to this method that it works best when you're not good at it. Once you get good you maintain your balance better and your legs don't have to work for you to stay on the line. I'm to the point where I practice on a completely slack "rodeo" line (which is very difficult and unforgiving) because I don't feel like normal ones exercise my legs enough.