Etymology
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Words related to inter-

*en 
Proto-Indo-European root meaning "in."

It forms all or part of: and; atoll; dysentery; embargo; embarrass; embryo; empire; employ; en- (1) "in; into;" en- (2) "near, at, in, on, within;" enclave; endo-; enema; engine; enoptomancy; enter; enteric; enteritis; entero-; entice; ento-; entrails; envoy; envy; episode; esoteric; imbroglio; immolate; immure; impede; impend; impetus; important; impostor; impresario; impromptu; in; in- (2) "into, in, on, upon;" inchoate; incite; increase; inculcate; incumbent; industry; indigence; inflict; ingenuous; ingest; inly; inmost; inn; innate; inner; innuendo; inoculate; insignia; instant; intaglio; inter-; interim; interior; intern; internal; intestine; intimate (adj.) "closely acquainted, very familiar;" intra-; intricate; intrinsic; intro-; introduce; introduction; introit; introspect; invert; mesentery.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit antara- "interior;" Greek en "in," eis "into," endon "within;" Latin in "in, into," intro "inward," intra "inside, within;" Old Irish in, Welsh yn, Old Church Slavonic on-, Old English in "in, into," inne "within, inside."
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entertain (v.)
Origin and meaning of entertain

late 15c., "to keep up, maintain, to keep (someone) in a certain frame of mind," from Old French entretenir "hold together, stick together, support" (12c.), from entre- "among" (from Latin inter; see inter-) + tenir "to hold" (from Latin tenere, from PIE root *ten- "to stretch").

Sense of "have a guest" is late 15c.; that of "gratify, amuse" is 1620s. Meaning "to allow (something) to consideration, take into the mind" (of opinions, notions, etc.) is 1610s. Related: Entertained; entertaining.

enterprise (n.)

early 15c., "an undertaking," formerly also enterprize, from Old French enterprise "an undertaking," noun use of fem. past participle of entreprendre "undertake, take in hand" (12c.), from entre- "between" (see entre-) + prendre "to take," contraction of prehendere "to catch hold of, seize" (from prae- "before," see pre-, + -hendere, from PIE root *ghend- "to seize, take"). Abstract sense of "adventurous disposition, readiness to undertake challenges, spirit of daring" is from late 15c.

intelligence (n.)
Origin and meaning of intelligence
late 14c., "the highest faculty of the mind, capacity for comprehending general truths;" c. 1400, "faculty of understanding, comprehension," from Old French intelligence (12c.) and directly from Latin intelligentia, intellegentia "understanding, knowledge, power of discerning; art, skill, taste," from intelligentem (nominative intelligens) "discerning, appreciative," present participle of intelligere "to understand, comprehend, come to know," from assimilated form of inter "between" (see inter-) + legere "choose, pick out, read," from PIE root *leg- (1) "to collect, gather," with derivatives meaning "to speak (to 'pick out words')."

Meaning "superior understanding, sagacity, quality of being intelligent" is from early 15c. Sense of "information received or imparted, news" first recorded mid-15c., especially "secret information from spies" (1580s). Meaning "a being endowed with understanding or intelligence" is late 14c. Intelligence quotient first recorded 1921 (see I.Q.).
entre- 
in words from French, corresponds to English enter-, which is itself from French entre "between, among" (11c.), from Latin inter (see inter-).
inter alia 
Latin, literally "amongst other things." from inter "among, between" (see inter-) + alia, neuter accusative plural of alius "(an)other" (see alias (adv.)). Latin for "among other persons" is inter alios.
interact (v.)
"act on each other, act reciprocally," 1805, from inter- + act (v.). Related: Interacted; interacting.
interamnian (adj.)
"between two rivers" (usually, if not exclusively, with reference to Mesopotamia), 1774, from Late Latin interamnius, from inter "between" (see inter-) + amnis "a river," a word perhaps of Celtic origin (see afanc).
interblend (v.)
"intermingle," 1823, from inter- + blend (v.). Related: Interblended; interblending.