koincidi

Do not try to split it. Ko- is not an Esperanto prefix - the language uses kun- for togetherness, as in kunlabori and kunveni - and incidi is not a verb here at all. Koincid- is one indivisible borrowed root. The three-part decomposition is completely real, but it happened in Latin: co- plus in plus cadere, to fall together onto the same point. Knowing that fixes the word in your memory and buys you nothing you can reuse, which is the normal deal with international roots.

The falling is worth keeping in mind, though, because it explains the range. Two lines that meet everywhere coincide in the geometric sense; two events on the same date coincide in the diary sense; two accounts of the same evening coincide when they agree. Esperanto uses the one verb for all three, and the cadere behind it is the same verb that gives incident (something that falls in), accident (something that falls to you) and cadence (the falling of a voice).

Coincidence, the noun, is two different words. Koincido is the fact of two things falling together - la koincido de la du datoj. Hazardo is chance itself, the luck of the thing, and it is what you reach for when the point is that nobody planned it: pura hazardo, sheer coincidence. English collapses both into coincidence and Esperanto does not, so choose deliberately: koincido describes the overlap, hazardo explains it.

Construction. The verb is intransitive and takes kun for the other party: koincidi kun io. Two subjects can share it without a preposition - la du datoj koincidas - and samtempe, at the same time, does the adverbial job when you want to say it about a verb instead.