Slim,
In considering the definition of Balanced Field Length I imagine the performance of a hypothetical aeroplane configured like a Cessna 337.(A push/pull twin-google it if you are not familiar). Except my imaginary aeroplane has powerful engines and is capable of taking off on one engine alone. There is no difficulty in keeping straight after engine failure at any speed.
Consider the following scenarios:
Accelerate Stop Distance- What if we accelerated all the way to a safe take off speed then failed an engine and stopped. The distance used on a runway would be relatively large. If we accelerated to 5 knots and stopped the distance would be a few meters. We could graph the distance for all intermediate speeds between 5 kts and safe take off speed.
Accelerate Go Distance- similarly if we take off from a standing start with a failed engine the distance required would be relatively long compared with normal take off.It follows that the lower the speed the engine is failed the greater the Accelerate Go Distance. Graph the result for intermediate failure speeds between 5kts and safe take off speed on the same graph.
The two lines on your graph will cross. This will be the speed and distance at which Accelerate Go and Accelerate Stop will be equal.
Hope this helps with the concept.
Greedy