You must eat something soon after waking otherwise you're running on empty and risking heart problems.
Is that really true? Just that pretty much everyone I've known who swims hard does it early before breakfast, as that's the usually the best time to pool swim.
I eat breakfast proper before riding to work, but then often have some milk or something once I get in (it is about 16 hilly miles), so I'd be in both the before and after camps.
I think whether you need to eat before is a matter of how long and how hard you exercise. If it's just a 45 minutes of easy riding on the flat, then I would be fine on an empty stomach, whereas an hour of hilly riding (or of riding very fast on the flat), I'd want something inside me.
Really though, what you need to do on this is listen to your body – if during exercise, you find yourself feeling weak and feeble suddenly because you run out of energy, that is your body telling you that it might make sense to eat something beforehand. The same afterwards – if you feel broken after rides, maybe your body needs something in afterwards too. Similarly, if you find you feel like vomiting on a post-breakfast ride, then eat less before the ride and more after or during it.
Joe