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Measurements of CP asymmetries in charmless three-body decays of B± mesons are reported using proton-proton collision data collected by the LHCb detector, corresponding to an integrated luminosity of 5.9 fb−1. The previously observed CP asymmetry in B±→π±K+K− decays is confirmed, and CP asymmetries are observed with a significance of more than five standard deviations in the B±→π±π+π− and B±→K±K+K− decays, while the CP asymmetry of B±→K±π+π− decays is confirmed to be compatible with zero. The distributions of these asymmetries are also studied as a function of the three-body phase space and suggest contributions from rescattering and resonance interference processes. An indication of the presence of the decays B±→π±χc0(1P) in both B±→π±π+π− and B±→π±K+K− decays is observed, as is CP violation involving these amplitudes.