We report results on studies of the e(+)e(-) annihilation into three-body Upsilon(nS)pi(+)pi(-) (n = 1, 2, 3) final states including measurements of cross sections and the full amplitude analysis. The cross sections measured at root s = 10.866 GeV and corrected for the initial state radiation are sigma(e(+)e(-) --> Upsilon(1S)pi(+)pi(-)) = (2.27 +/- 0.12 +/- 0.14) pb, sigma(e(+)e(-) --> Upsilon(2S)pi(+)pi(-)) = (4.07 +/- 0.16 +/- 0.45) pb, and sigma(e(+)e(-) --> Upsilon(3S)pi(+)pi(-)) = (1.46 +/- 0.09 +/- 0.16) pb. Amplitude analysis of the three-body Upsilon(nS)pi(+)pi(-) final states strongly favors I-G(J(P)) = 1(+)(1(+)) quantum-number assignments for the two bottomonium-like Z(b)(+/-) states, recently observed in the Upsilon(nS)pi(+) and h(b)(mP)pi(+)(m = 1, 2) decay channels. The results are obtained with a 121.4 fb(-1) data sample collected with the Belle detector at the KEKB asymmetric-energy e(+)e(-) collider.