Etymology
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Words related to egest

ex- 
word-forming element, in English meaning usually "out of, from," but also "upwards, completely, deprive of, without," and "former;" from Latin ex "out of, from within; from which time, since; according to; in regard to," from PIE *eghs "out" (source also of Gaulish ex-, Old Irish ess-, Old Church Slavonic izu, Russian iz). In some cases also from Greek cognate ex, ek. PIE *eghs had comparative form *eks-tero and superlative *eks-t(e)r-emo-. Often reduced to e- before -b-, -d-, -g-, consonantal -i-, -l-, -m-, -n-, -v- (as in elude, emerge, evaporate, etc.).
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gest (n.)

"famous deed, exploit," more commonly "story of great deeds, tale of adventure," c. 1300, from Old French geste, jeste "action, exploit, romance, history" (of celebrated people or actions), from Medieval Latin gesta "actions, exploits, deeds, achievements," noun use of neuter plural of Latin gestus, past participle of gerere "to carry on, wage, perform," which de Vaan says is considered to be from the same root as agere "to set in motion, drive forward, do, perform" (from PIE root *ag- "to drive, draw out or forth, move"). Now only as a deliberate archaism. Jest (n.) is the same word, with a decayed sense.

ingest (v.)
1610s, "to take in as food," from Latin ingestus, past participle of ingerere "to throw in, pour in, heap upon," from in- "into" (from PIE root *en "in") + gerere "to carry" (see gest). Related: Ingested; ingesting.
egestion (n.)
"act of voiding the refuse of digestion," early 15c., from Latin egestionem (nominative egestio), noun of action from past participle stem of egerere "to discharge" (see egest).