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English word rin comes from Proto-Indo-European *ren-, and later Proto-Germanic *rinnaną ((of liquid) to flow. To run, to rush (move quickly).)
*ren- (Proto-Indo-European)
to rise and sink, bob, to rise; to sink
*rinnaną (Proto-Germanic)
(of liquid) to flow. To run, to rush (move quickly).
rinna (Old Norse)
(intransitive) to melt. (intransitive) to run (move quickly). (intransitive, of liquid) to flow, run.
rinnan (Old English)
To run.
irnan (Old English)
rinnen (Middle English)
rin (English)
(ambitransitive, dialectal) To run.