We perform amplitude analyses of the decays B0→K+K-KS0, B+→K+K-K+, and B+→KS0KS0K+, and measure CP-violating parameters and partial branching fractions. The results are based on a data sample of approximately 470×106 BB̄ decays, collected with the BABAR detector at the PEP-II asymmetric-energy B factory at the SLAC National Accelerator Laboratory. For B+→K+K-K+, we find a direct CP asymmetry in B+→(1020)K+of ACP=(12.8±4.4±1.3)%, which differs from zero by 2.8σ. For B0→K+K-KS0, we measure the CP-violating phase βeff((1020)KS0)=(21±6±2)°. For B+→KS0KS0K+, we measure an overall direct CP asymmetry of ACP=(4-5+4±2)%. We also perform an angular-moment analysis of the three channels and determine that the fX(1500) state can be described well by the sum of the resonances f0(1500), f2′(1525), and f0(1710). © 2012 American Physical Society.