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