Etymology
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Words related to Reich

*reg- 

Proto-Indo-European root meaning "move in a straight line," with derivatives meaning "to direct in a straight line," thus "to lead, rule."

It forms all or part of: abrogate; address; adroit; Alaric; alert; anorectic; anorexia; arrogant; arrogate; bishopric; correct; corvee; derecho; derogate; derogatory; Dietrich; direct; dress; eldritch; erect; ergo; Eric; Frederick; Henry; incorrigible; interregnum; interrogate; maharajah; Maratha; prerogative; prorogue; rack (n.1) "frame with bars;" rail (n.1) "horizontal bar passing from one post or support to another;" Raj; rajah; rake (n.1) "toothed tool for drawing or scraping things together;" rake (n.2) "debauchee; idle, dissolute person;" rakish; rank (adj.) "corrupt, loathsome, foul;" real (n.) "small Spanish silver coin;" realm; reck; reckless; reckon; rectangle; rectify; rectilinear; rectitude; recto; recto-; rector; rectum; regal; regent; regicide; regime; regimen; regiment; region; regular; regulate; Regulus; Reich; reign; resurgent; rex; rich; right; Risorgimento; rogation; royal; rule; sord; source; subrogate; subrogation; surge; surrogate; viceroy.

It is the hypothetical source of/evidence for its existence is provided by:

Sanskrit raj- "a king, a leader," rjyati "he stretches himself," riag "torture" (by racking); Avestan razeyeiti "directs," raštva- "directed, arranged, straight;" Persian rahst "right, correct;" Latin regere "to rule, direct, lead, govern," rex (genitive regis) "king," rectus "right, correct;" Greek oregein "to reach, extend;" Old Irish ri, Gaelic righ "a king," Gaulish -rix "a king" (in personal names, such as Vircingetorix), Old Irish rigim "to stretch out;" Gothic reiks "a leader," raihts "straight, right;" Lithuanian raižytis "to stretch oneself;" Old English rice "kingdom," -ric "king," rice "rich, powerful," riht "correct;" Gothic raihts, Old High German recht, Old Swedish reht, Old Norse rettr "correct."

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third (adj., n.)

"next in order after the second; an ordinal numeral; being one of three equal parts into which a whole is regarded as divided;" late Old English metathesis of þridda, from Proto-Germanic *thridja- (source also of Old Frisian thredda, Old Saxon thriddio, Middle Low German drudde, Dutch derde, Old High German dritto, German dritte, Old Norse þriðe, Danish tredie, Swedish tredje, Gothic þridja), from PIE *tri-tyo- (source also of Sanskrit trtiyas, Avestan thritya, Greek tritos, Latin tertius (source of Italian terzo, Spanish tercio, French tiers), Old Church Slavonic tretiji, Lithuanian trečias, Old Irish triss, Welsh tryde), suffixed form of root *trei- (see three).

Metathesis of thrid into third is attested from c. 950 in Northumbrian (compare wright), but overall thrid was prevalent up to 16c. The noun meaning "third part of anything" is recorded from late 14c. Third rail in electric railway sense is recorded from 1890. Third World War as a possibility first recorded 1947. Third-rate "of poor quality" is from 1814, ultimately from classification of ships (1640s); third class in railway travel is from 1839. Third Reich (1930) is a partial translation of German drittes Reich (1923). Third party in law, insurance, etc., is from 1818.

Reichstag (n.)

1867 as the name of parliaments in central European nations, especially "the German imperial parliament" (1871-1918), from German Reichstag, from Reich "state, realm" (see Reich) + Tag "assembly," literally "day" (see day).

Earliest use in English is in reference to the chief deliberative body of the North German Confederacy. Also later as the name of the building (opened in 1894) in Berlin in which the imperial parliament met; hence Reichstag Fire (which took place Feb. 27, 1933) as symbolic of a disruptive act engineered to facilitate the rise of a party to power.

rich (adj.)

Old English rice "strong, powerful; great, mighty; of high rank" (senses now obsolete), in later Old English "wealthy;" from Proto-Germanic *rikijaz (source also of Old Norse rikr, Swedish rik, Danish rig, Old Frisian rike "wealthy, mighty," Dutch rijk, Old High German rihhi "ruler, powerful, rich," German reich "rich," Gothic reiks "ruler, powerful, rich"), borrowed from a Celtic source akin to Gaulish *rix, Old Irish ri (genitive rig) "king," from Proto-Celtic *rix, from PIE root *reg- "move in a straight line," with derivatives meaning "to direct in a straight line," thus "to lead, rule" (compare rex).

The form of the word was influenced in Middle English by Old French riche "wealthy, magnificent, sumptuous," which is, with Spanish rico, Italian ricco, from Frankish *riki "powerful," or some other cognate Germanic word. Old English also had a noun, rice "rule, reign, power, might; authority; empire" (compare Reich). The evolution of the word reflects a connection between wealth and power in the ancient world, though the "power" sense seems to be the oldest.

In transferred and extended senses from c. 1200. The meaning "magnificent" is from c. 1200; that of "of great value or worth" is from mid-13c. Of food and colors, "having an abundance of a characteristic quality that pleases the senses," from early 14c.; of sounds, from 1590s; of soils from 1570s. Sense of "entertaining, amusing" is recorded from 1760. The noun meaning "the wealthy" was in Old English.

English once had a related verb rixle "have domination, rule," from Old English rixian "to rule."

Theodoric 
masc. proper name, from Late Latin Theodoricus, from Gothic, literally "ruler of the people," from Gothic þiuda "people" (see Teutonic) + *reiks "ruler" (see Reich). For spelling, see Theobald. The French form of the name, via the Franks, is Thierry.