As i see it.
The dog has no clue where or how far.. The most he can go is half way before he heads out so...
But if the dog had a problem solving brain he would stop at the farthest point eg the middle
so ask yourself this.
What's required to work out how far in? Eg a compass. eg how far have you traveled...
or rough calculations how fast roughly does the dog run, how big is the forest? Distance/time = roughly where dog is if traveling in a straight line.. i personaly woukd just use a gps 