A bstract A data sample collected with the LHCb detector corresponding to an integrated luminosity of 9 fb − 1 is used to measure eleven CP violation observables in B ± → Dh ± decays, where h is either a kaon or a pion. The neutral D meson decay is reconstructed in the three-body final states: K ± π ∓ π 0 ; π + π − π 0 ; K + K − π 0 and the suppressed π ± K ∓ π 0 combination. The mode where a large CP asymmetry is expected, B ± → [ π ± K ∓ π 0 ] D K ± , is observed with a significance greater than seven standard deviations. The ratio of the partial width of this mode relative to that of the favoured mode, B ± → [ K ± π ∓ π 0 ] D K ± , is R ADS( K ) = (1 . 27 ± 0 . 16 ± 0 . 02) × 10 − 2 . Evidence for a large CP asymmetry is also seen: A ADS( K ) = −0 . 38 ± 0 . 12 ± 0 . 02. Constraints on the CKM angle γ are calculated from the eleven reported observables.