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Abstract A method is proposed to measure the photon polarisation parameter $$\lambda _{\gamma }$$ λγ in $${{b}} \!\rightarrow {s} {\gamma } $$ b→sγ transitions using an amplitude analysis of $$B\!\rightarrow K\pi \pi {\gamma } $$ B→Kππγ decays. Simplified models of the $${K} {\pi } {\pi } $$ Kππ system are used to simulate $${{{B} ^+}} \!\rightarrow {{K} ^+} {{\pi } ^-} {{\pi } ^+} {\gamma } $$ B+→K+π-π+γ and $${{B} ^0} \!\rightarrow {{K} ^+} {{\pi } ^-} {{\pi } ^0} {\gamma } $$ B0→K+π-π0γ decays, validate the amplitude analysis method, and demonstrate the feasibility of a measurement of the $$\lambda _{\gamma }$$ λγ parameter irrespective of the model parameters. Similar sensitivities to $$\lambda _{\gamma }$$ λγ are obtained with both the charged and neutral hadronic systems. In the absence of any background and distortion due to experimental effects, the statistical uncertainty expected from an analysis of $${{{B} ^+}} \!\rightarrow {{K} ^+} {{\pi } ^-} {{\pi } ^+} {\gamma } $$ B+→K+π-π+γ decays in an LHCb data set corresponding to an integrated luminosity of 9 $$\,\hbox {fb}^{-1}$$ fb-1 is estimated to be 0.009. A similar measurement using $${{B} ^0} \!\rightarrow {{K} ^+} {{\pi } ^-} {{\pi } ^0} {\gamma } $$ B0→K+π-π0γ decays in a Belle II data sample corresponding to an integrated luminosity of 5 $$\hbox {\,ab}^{-1}$$ ab-1 would lead to a statistical uncertainty of 0.018.