From Middle English feyen, feien, from Old English fēġan ("to join, unite"), from Proto-Germanic *fōgijaną ("to join"), from *fōgō, from Proto-Indo-European *paḱ- ("to fasten, place"). Akin to Old Frisian fōgia ("to join"), Old Saxon fōgian ("to join"), Middle Low German fögen ("to join, add"), Dutch voegen ("to add, place"), Old High German fuogen ("to connect") (German fügen ("to connect")), Old English fōn ("to catch"). More at fang.
From Middle English fegien, fæien, from Old Norse fægja ("to cleanse, polish"), from Proto-Germanic *fēgijaną ("to decorate, make beautiful"), from Proto-Indo-European *pōḱ-, *pēḱ-. Cognate with Swedish feja ("to sweep"), Danish feje ("to sweep"), German fegen ("to cleanse, scour, sweep"), Dutch vegen ("to sweep, strike"). More at feague, fake, fair.
Middle English faie, fei, from Middle French feie, fee. More at fairy.