Etymology
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shoe (n.)
Old English scoh "shoe," from Proto-Germanic *skokhaz (source also of Old Norse skor, Danish and Swedish sko, Old Frisian skoch, Old Saxon skoh, Middle Dutch scoe, Dutch schoen, Old High German scuoh, German Schuh, Gothic skoh). No known cognates outside Germanic, unless it somehow is connected with PIE root *skeu- "cover" (source also of second element in Latin ob-scurus).

Old plural form shoon lasted until 16c. Meaning "metal plate to protect a horse's hoof" is attested from late 14c. Distinction between shoe and boot (n.) is attested from c. 1400. To stand in someone's shoes "see things from his or her point of view" is attested from 1767. Old shoe as a type of something worthless is attested from late 14c.

Shoes tied to the fender of a newlywed couple's car preserves the old custom (mentioned from 1540s) of throwing an old shoe at or after someone to wish them luck. Perhaps the association is with dirtiness, on the "muck is luck" theory.
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shoe (v.)
Old English scogan "to shoe," from the root of shoe (n.). In reference to horses from c. 1200. Related: Shoed; shoeing.
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running (adj.)

"that runs, capable of moving quickly," late 14c., rennynge, present-participle adjective from run (v.), replacing earlier erninde, from Old English eornende. The meaning "rapid, hasty, done on the run" is from c. 1300. The sense of "continuous, carried on continually" is from late 15c.

Running-jump is from 1914. A running-mate (1865) originally was a horse entered in a race to set the pace for another from the same stable who was intended to win; U.S. "vice-presidential candidate" sense is recorded from 1888. Running-board is attested by 1817 in reference to a narrow gangway on either side of a ship or boat; extended by 1907 to the footboards of cars and trucks. 

Running dog is recorded by 1937, from Chinese and later North Korean communist phrases used to describe supposed imperialist lackeys, such as Mandarin zou gou "running dog," on the notion of a dog that runs at its master's command.

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running (n.)

Old English ærning, "act of one who or that which runs, rapid motion on foot," verbal noun from run (v.). Of a ship, "the action of sailing," 1680s.

Colloquial phrases in (or out) of the running "among (or not among) the lead competitors, competing (or not competing) in a race" (1863) is a metaphor from horse racing, where make the running "set the pace" is recorded from 1837; hence "likely to succeed." Running-shoe is from 1884.

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snow-shoe (n.)
also snowshoe, 1670s, from snow (n.) + shoe (n.). Related: Snowshoes.
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long-running (adj.)
1943, of theatrical productions, from long (adv.) + present participle of run (v.). Related: Longest-running.
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shoe-shine (adj.)
1911, from shoe (n.) + shine (n.). One who shines shoes for money was a shoeblacker (1755).
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shoe-string (n.)
1610s, from shoe (n.) + string (n.). As figurative for "a small amount" it is recorded from 1882; as a type of necktie, from 1903; as a style of cooked potatoes from 1906.
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overshoe (n.)

also over-shoe, "a shoe worn over another," especially "an outer waterproof shoe," 1829, from over- + shoe (n.). Related: Overshoes.

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shoelace (n.)

also shoe-lace, 1640s, from shoe (n.) + lace (n.).An older word for the thong or lace of a shoe or boot was Middle English sho-thong, Old English scoh-þwang.

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