  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 $B^\pm\to Dh^\pm$ decays, where $h$ is either a kaon or a pion. The neutral
$D$ meson decay is reconstructed in the three-body final states:
$K^\pm\pi^\mp\pi^0$}; $\pi^+\pi^-\pi^0$; $K^+K^-\pi^0$ and the suppressed
$\pi^\pm K^\mp\pi^0$ combination. The mode where a large $CP$ asymmetry is
expected, $B^\pm\to [\pi^\pm K^\mp\pi^0]_DK^\pm$, 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, $B^\pm\to
[K^\pm\pi^\mp\pi^0]_D K^\pm$, is $R_{{\rm ADS}(K)} = (1.27\pm0.16\pm0.02)\times
10^{-2}$. Evidence for a large $CP$ asymmetry is also seen: $A_{{\rm ADS}(K)} =
-0.38\pm0.12\pm0.02$. Constraints on the CKM angle $\gamma$ are calculated from
the eleven reported observables.
