Notice the diagram. If a point lies in the YELLOW region, it will be closer to the midpoint of the diagonal than A or C; and if the point lies in the GREEN region, it will be closer to A or C than the midpoint of the diagonal. Let x be the length of each side of the square. So the required probability istotal area of the GREEN region / total area of the square = [{(1/2)*(x/2)*(x/2)}*2] / x*x = 1/8