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HD 209458b is an exoplanet with an upper atmosphere undergoing blow-off escape that has mainly been studied using measurements of the Ly-alpha absorption. Recently, high-resolution measurements of absorption in the He I triplet line at 10830 angstroms of several exoplanets (including HD 209458b) have been reported, creating a new opportunity to probe escaping atmospheres. We aim to better understand the atmospheric regions of HD 209458b from where the escape originates. We developed a 1D hydrodynamic model with spherical symmetry for the HD 209458 b thermosphere coupled with a non-local thermodynamic model for the population of the He triplet state. In addition, we performed high-resolution radiative transfer calculations of synthetic spectra for the He triplet lines and compared them with the measured absorption spectrum in order to retrieve information about the atmospheric parameters. We find that the measured spectrum constrains the [H]/[H$^{+}$] transition altitude occurring in the range of 1.2 to 1.9Rp. H is almost fully ionised at altitudes above 2.9Rp. We also find that the X-ray and EUV absorption takes place at effective radii from 1.16 to 1.30Rp, and that the He triplet peak density occurs at altitudes from 1.04 to 1.60Rp. Additionally, the averaged mmw is confined to the 0.61-0.73 g/mole interval, and the thermospheric H/He ratio should be larger than 90/10, and most likely approximately 98/2. We also provide a one-to-one relationship between mass-loss rate and temperature. Based on the energy-limited escape approach and assuming heating efficiencies of 0.1-0.2, we find a mass-loss rate in the range of (0.42-1.00)$\times 10^{11}$ g/s and a corresponding temperature range of 7125 to 8125K. The analysis of the measured He triplet absorption spectrum significantly constrains the thermospheric structure of HD 209458b and advances our knowledge of its escaping atmosphere.