An Etymological Dictionary of the German Language/Fächer
Fächer, earlier also Fächel, m., ‘fan,’ ModHG. only; the derivation is uncertain; perhaps a diminutive of MidHG. vach, ‘veil.’ Yet the suspicion that the word was borrowed is not unfounded, since MidHG. foche, focher, ‘fan,’ point to Lat. focarius, foculare (from focus). The change of a to o may be due to LG. (comp. Aberglaube, Adebar), as in anfachen, from Lat. focare.