Amplitude models are constructed to describe the resonance structure of D0→K−π+π+π− and D0→K+π−π−π+ decays using pp collision data collected at centre-of-mass energies of 7 and 8 TeV with the LHCb experiment, corresponding to an integrated luminosity of 3.0 fb−1. The largest contributions to both decay amplitudes are found to come from axial resonances, with decay modes D0→a1(1260)+K− and D0→K1(1270/1400)+π− being prominent in D0→K−π+π+π− and D0→K+π−π−π+, respectively. Precise measurements of the lineshape parameters and couplings of the a1(1260)+, K1(1270)− and K(1460)− resonances are made, and a quasi model-independent study of the K(1460)− resonance is performed. The coherence factor of the decays is calculated from the amplitude models to be RK3π=0.459±0.010(stat)±0.012(syst)±0.020(model), which is consistent with direct measurements. These models will be useful in future measurements of the unitary-triangle angle γ and studies of charm mixing and CP violation.