Measurements of $CP$ asymmetries in charmless three-body decays of $B^{\pm }$ mesons are reported using proton-proton collision data collected by the LHCb detector, corresponding to an integrated luminosity of $5.9{\mathrm{fb}}^{-1}$. The previously observed $CP$ asymmetry in $B^{\pm }\to {\pi }^{\pm }{K}^{+}{K}^{-}$ decays is confirmed, and $CP$ asymmetries are observed with a significance of more than five standard deviations in the $B^{\pm }\to {\pi }^{\pm }{\pi }^{+}{\pi }^{-}$ and $B^{\pm }\to K^{\pm }{K}^{+}{K}^{-}$ decays, while the $CP$ asymmetry of $B^{\pm }\to K^{\pm }{\pi }^{+}{\pi }^{-}$ 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 $B^{\pm }\to {\pi }^{\pm }{\chi }_{c0}\left(1P\right)$ in both $B^{\pm }\to {\pi }^{\pm }{\pi }^{+}{\pi }^{-}$ and $B^{\pm }\to {\pi }^{\pm }{K}^{+}{K}^{-}$ decays is observed, as is $CP$ violation involving these amplitudes.