I would challenge that you need to eat before you workout. In fact good ol' Arnie advocated fasting beforehand and even did a video back in the 70's about it called "Stay Hungry". It was still in video stores not too long ago so you might find it on the internet somewhere. According to modern nutrition and bodybuilding theory though I agree it says you should eat carbs within 30 minutes of finishing a workout and then a protein meal soon after so that the protein is in your body the next day when the body is supposed to use it to build the extra muscle tissue.
In my vegetarian research though I've discovered some of the basis for the process described above to be flawed. Because it appears to be based on the old rubbish that you need to eat a 'complete' protein for the body to used that complete protein to build more muscle. This is rubbish. Our body creates new tissue including muscle mass from it's stores, metabolising the protein it needs from various places to make up just the right mix for the body part it needs to create. We don't need to eat flesh to make flesh! As long as we have a varied diet and are getting full nutrition over weeks and months it is stored for use when we need it. Fasting encourages that process even more because it only uses the stores no matter what it is doing.
The only question to answer as to whether the body can build muslce while fasting is does it need the energy elsewhere for detoxification or repairs.