intersect v.

1550s, "act or fact of crossing," from French intersection (14c.) and directly from Latin intersectionem (nominative intersectio) "a cutting asunder, intersection," noun of action from past-participle stem of intersecare "intersect, cut asunder," from inter- "between" (see inter-) + secare "to cut" (from PIE root *sek- "to cut"). In English originally a term in geometry; meaning "crossroads, a place of crossing" is from 1864. Related: Intersectional.

word-forming element used freely in English, "between, among, during," from Latin inter (prep., adv.) "among, between, betwixt, in the midst of" (also used extensively as a prefix), from PIE *enter "between, among" (source also of Sanskrit antar, Old Persian antar "among, between," Greek entera (plural) "intestines," Old Irish eter, Old Welsh ithr "among, between," Gothic undar, Old English under "under"), a comparative of root *en "in."

A living prefix in English from 15c. and used with Germanic as well as Latinate words. Spelled entre- in French; most words borrowed into English in that form were re-spelled 16c. to conform with Latin except entertain, enterprise. In Latin, spelling shifted to intel- before -l-, hence intelligence, etc.

Proto-Indo-European root meaning "to cut." It forms all or part of: bisect; dissect; hacksaw; insect; intersect; resect; saw (n.1) "cutting tool;" Saxon; scythe; secant; secateurs; sect; section; sector; sedge; segment; skin; skinflint; skinny; transect.

It is the hypothetical source of/evidence for its existence is provided by: Hittite shakk- "to know, pay attention to;" Latin secare "to cut," sectio "a cutting, cutting off, division;" Old Church Slavonic seko, sešti "to cut," sečivo "ax, hatchet," Russian seč' "to cut to pieces;" Lithuanian įsėkti "to engrave, carve;" Albanian šate "mattock;" Old Saxon segasna, Old English sigðe "scythe;" Old English secg "sword," seax "knife, short sword;" Old Irish doescim "I cut."