OMCC 2002 Problem 6

A path from $(0,0)$ to $(n,n)$ on the lattice is made up of unit moves upward or rightward. It is balanced if the sum of the x-coordinates of its $2n+1$ vertices equals the sum of their y-coordinates. Show that a balanced path divides the square with vertices $(0,0)$ , $(n,0)$ , $(n,n)$ , $(0,n)$ into two parts with equal area.