Achernar, the closest and brightest classical Be star, presents rotational flattening, gravity darkening, occasional emission lines due to a gaseous disk, and an extended polar wind. It is also a member of a close binary system with an early A-type dwarf companion. We aim to determine the orbital parameters of the Achernar system and to estimate the physical properties of the components. Methods. We monitored the relative position of Achernar B using a broad range of high angular resolution instruments of the VLT/VLTI (VISIR, NACO, SPHERE, AMBER, PIONIER, GRAVITY, and MATISSE) over a period of 13 years (2006-2019). These astrometric observations are complemented with a series of ~750 optical spectra for the period from 2003 to 2016. We determine that Achernar B orbits the primary Be star on a seven-year period, eccentric orbit (e=0.7255+/-0.0014) which brings the two stars within 2au at periastron. The mass of the Be star is found to be mA=6.0+/-0.6M_{sun} for a secondary mass of mB=2.0+/-0.1M{sun} (the latter was estimated from modeling). We find a good agreement of the parameters of Achernar A with the evolutionary model of a critically rotating star of 6.4M{sun}_ at an age of 63Ma. The equatorial plane of the Be star and the orbital plane of the companion exhibit a relative inclination of 30{deg}. We also identify a resolved comoving low-mass star, which leads us to propose that Achernar is a member of the Tucana-Horologium moving group. The proximity of Achernar makes this star a precious benchmark for stellar evolution models of fast rotators and intermediate mass binaries. Achernar A is presently in a short-lived phase of its evolution following the turn-off, during which its geometrical flattening ratio is the most extreme. Considering the orbital parameters, no significant interaction occurred between the two components, demonstrating that Be stars may form through a direct, single-star evolution path without mass transfer. Since component A will enter the instability strip in a few hundred thousand years, Achernar appears to be a promising progenitor of the Cepheid binary systems.
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