Etymology
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Words related to *reig-

reach (v.)

Middle English rēchen, from Old English ræcan, reccan "to reach out, stretch or extend outward, hold forth, extend in continuity or scope," also "to succeed in touching, succeed in striking;" also "to address, speak to," also "to offer, present, give, grant."

This is proposed to be from Proto-West Germanic *raikejanan "stretch out the hand" (source also of Old Frisian reka "to give, pay," Middle Dutch reken, reiken, Old High German reihhen, reichen "give, reach out, get," Dutch reiken,  German reichen "to reach, to pass, to hand, to give; to be sufficient"), from Proto-Germanic *raikijanau, which is probably from PIE root *reig- "to stretch, stretch out, be stretched; be stiff."

Sometimes 16c. spelled retch. As "to hand (someone something), give" from c. 1300. The meaning "arrive at, succeed in getting to" is early 14c.; that of "succeed in influencing" is from 1660s. Related: Reached; reaching. Shakespeare uses the now-obsolete past tense form raught (Old English ræhte).

Colloquial reach-me-down "ready-made" (of clothes) is recorded from 1862, from notion of being on the rack in a finished state.

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rigid (adj.)

"stiff, unyielding, inflexible, firm, not easily bent," early 15c., from Latin rigidus "hard, stiff, rough, severe," from rigēre "be stiff" (from PIE root *reig- "stretch; be stretched; be stiff"). Related: Rigidly. As a verb, "to make rigid," rigidize is attested by 1944 in U.S. military and commercial use in reference to metals; earlier was rigidify (1842), rigidification.

rigidity (n.)

1620s, "stiffness, inflexibility," especially in mechanics, "resistance to change of form;" 1620s, from Latin rigiditas "stiffness," from rigidus "hard, stiff, rough, severe" (see rigid). By 1650s as "strictness, severity," but rigidity tends to be used of physical stiffness, while rigor is more active or moral. Rigidness (1640s) "perhaps holds a middle position" [Century Dictionary].

Rigidity is directly opposed to flexibility, and only indirectly to malleability and ductility, which depend chiefly on relations between the tenacity, the rigidity, and the limit of elasticity. [Century Dictionary]
rigor (n.)

late 14c., rigour, "harshness, severity in dealing with persons; force; cruelty," from Old French rigor "strength, hardness" (13c., Modern French rigueur), from Latin rigorem (nominative rigor) "numbness, stiffness, hardness, firmness; roughness, rudeness," from rigēre "be stiff" (from PIE root *reig- "stretch; be stretched; be stiff").

Also, in medieval medicine, "a sudden chill" (c. 1400). From early 15c. as "exactness, strictness without indulgence" (of discipline, the law, etc.). Compare rigidity. Rigorism "rigidity in principles or practice" (originally religious) is from 1704. Rigidist is by 1716.