The reaction γ + p → K + + Σ + π was used to determine the invariant mass distributions or "line shapes" of the Σ + π − , Σ − π + and Σ 0 π 0 final states, from threshold at 1328 MeV/c 2 through the mass range of the Λ(1405) and the Λ(1520). The measurements were made with the CLAS system at Jefferson Lab using tagged real photons, for center-of-mass energies 1.95 < W < 2.85 GeV. The three mass distributions differ strongly in the vicinity of the I = 0 Λ(1405), indicating the presence of substantial I = 1 strength in the reaction. Background contributions to the data from the Σ 0 (1385) and from K * Σ production were studied and shown to have negligible influence. To separate the isospin amplitudes, Breit-Wigner model fits were made that included channel-coupling distortions due to the NK threshold. A best fit to all the data was obtained after including a phenomenological I = 1, J P = 1/2 − amplitude with a centroid at 1394 ± 20 MeV/c 2 and a second I = 1 amplitude at 1413 ± 10 MeV/c 2 . The centroid of the I = 0 Λ(1405) strength was found at the Σπ threshold, with the observed shape determined largely by channel-coupling, leading to an apparent overall peak near 1405 MeV/c 2 .