Aha: Choose n in Z+ and for each fixed k in 0 <= k < n find the product of the complex numbers P(k; n) = (zeta_n)^j - (zeta_n)^k, j in {0, 1, ..., n-1} - {k} over the range 0 <= k < n. Let Q = Product over k=0,1,..., n-1 of P(k: n), where zeta_n denotes exp(2*π*i/n). Then clearly |Q| = (f(n))^n(n-1) where f(n) is the geometric mean of chord-lengths of all chords of the unit circle containing say the point 1. The rest of the proof is left as an exercise. —Dan ----- Thanks everyone, Tom, Warut, James. The significance of those integrals is that their summing to zero implies: [The geometric mean of the chord-lengths of all the chords containing some given point of the unit circle] = 1. Is there maybe a direct way to see that? -----