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Old English fang "prey, spoils, plunder, booty; a seizing or taking," from gefangen, past participle of fon "seize, take, capture," from Proto-Germanic *fango- (cf. Old Frisian fangia, Middle Dutch and Dutch vangen, Old Norse fanga, German fangen, Gothic fahan), from PIE root *pag- "to make firm, fix;" connected to Latin pax (genitive pacis) "peace" (see pact).
The sense of "canine tooth" (1550s) probably developed from Old English fengtoð, literally "catching- or grasping-tooth." Transferred to the venom tooth of a serpent, etc., by 1800.