bid (v.)youdaoicibaDictYouDict[bid 词源字典]
probably an early Middle English mutual influence or confusion of two old words: The sense in bid farewell is from Old English biddan "to ask, entreat, beg, pray, beseech; order" (class V strong verb, past tense bæd, past participle beden), from Proto-Germanic *bidjan "to pray, entreat" (source also of German bitten "to ask," attested in Old High German from 8c., also Old Saxon bidjan, Old Frisian bidda, Old Norse biðja, Gothic bidjan). This, according to Kluge and Watkins, is from a PIE root *gwhedh- "to ask, pray" (see bead (n.)).

To bid at an auction, meanwhile, is from Old English beodan "offer, proclaim" (class II strong verb; past tense bead, past participle boden), from Proto-Germanic *beudan "to stretch out, reach out, offer, present," (source also of German bieten "to offer," Old High German biatan, also Old Saxon biodan, Old Frisian biada, Old Norse bjoða, Gothic anabiudan "to command"). This is from PIE root *bheudh- "to be aware, make aware" (see bode (v.)). As a noun, 1788, from the verb.[bid etymology, bid origin, 英语词源]
reach (v.)youdaoicibaDictYouDict
Old English ræcan, reccan "reach out, stretch out, extend, hold forth," also "succeed in touching, succeed in striking; address, speak to," also "offer, present, give, grant," from West Germanic *raikjan "stretch out the hand" (cognates: Old Frisian reka, Middle Dutch reiken, Dutch reiken, Old High German and German reichen), from Proto-Germanic *raikijanau, perhaps from PIE root *reig- "to stretch out" (cognates: Sanskrit rjyati "he stretches himself," riag "torture" (by racking); Greek oregein "to reach, extend;" Lithuanian raižius "to stretch oneself;" Old Irish rigim "I stretch").

Shakespeare uses the now-obsolete past tense form raught (Old English ræhte). Meaning "arrive at" is early 14c.; that of "succeed in influencing" is from 1660s. Related: Reached; reaching. Reach-me-down "ready-made" (of clothes) is recorded from 1862, from notion of being on the rack in a finished state.