I think all they were trying to show with the calculation of average intensity is that conservation of energy still applies, so the average intensity is equal to the sum of the intensity of the two individual waves. To find this average they determine the amplitude of the wave at two separate points - one where the waves are 60 degrees apart and another where they are 60+180 = 240 degrees apart. Mathematically they do this by first adding the two vectors (to get intensity = sqrt(3)A^2 and then by subtracting one from the other (to get intensity = A^2. Then they take the average of the two. This method works for any value of phase angle - for any point that has intensity of kA^2 there is another point that has intensity (4-k^2)A^2, for an average between the two points of 2A^2.