Can you strength-train enough for cycling by just using your own body weight, i.e. pushups, situps, etc, or do you need to use actual weights?

I used to ride quite a bit, even did a century ride at one point (El Tour de Tucson), but then I got into an accident and broke a few bones. I've been off the bike for about 2 years and a half and gained a bit of weight - not really too bad, but more than I am comfortable with. I started riding again this month in the beautiful Arizona desert, and I forgot how fun this sport is. I need to get in shape and thought some sort of strength-training would help with some base miles. I'm starting off at ground zero, so I'm open to suggestions