eintreten
to enter
To enter a room, building, or area.
Als er ins Zimmer eintrat, wurden alle still.
When he entered the room, everyone fell silent.

to enter / occur
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to enter
To enter a room, building, or area.
Als er ins Zimmer eintrat, wurden alle still.
When he entered the room, everyone fell silent.

'Eintreten' (to enter / occur) is a separable verb. In a main clause, the prefix 'ein' goes at the end. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Bitte treten Sie ein!
Please come in!
The perfect tense of 'eintreten' (to enter / occur) is always formed with the auxiliary verb 'sein' (to be), because it describes movement or a change of state. So the form is 'ist eingetreten' (has entered / has occurred).
Eine unerwartete Besserung ist eingetreten.
An unexpected improvement has occurred.
The verb is made up of 'ein-' (in, into) and 'treten' (to step). The literal meaning 'hinein-treten' (to step inside) helps you understand the meanings 'to enter a room' and 'to become a member'.
Der Gast trat aus der Kälte in das warme Haus ein.
The guest stepped out of the cold and into the warm house.
To express that someone becomes a member, you use the construction 'in etwas (Akkusativ) eintreten' (to join something). Typical nouns are 'Verein' (club), 'Partei' (party), or 'Organisation' (organization).
Letztes Jahr ist er in einen Sportverein eingetreten.
Last year he joined a sports club.
When someone stands up for something, the fixed phrase is 'für etwas (Akkusativ) eintreten' (to advocate for something). You can advocate for an idea, a right, or a person.
Die Anwältin ist für die Rechte ihrer Mandantin eingetreten.
The lawyer advocated for her client's rights.
In the meaning 'geschehen' (to happen) or 'passieren' (to occur), 'eintreten' (to occur) is used without an object. It often describes the beginning of a new state or event and is frequently used in formal language.
Die Dunkelheit trat langsam ein.
Darkness slowly set in.
In the meaning 'etwas mit dem Fuß aufstoßen' (to kick something open), 'eintreten' (to kick in) is used with a direct accusative object and without a preposition. Most often, the object is a door.
Die Polizei musste die Wohnungstür eintreten.
The police had to kick in the apartment door.
to come
'kommen' and 'eintreten' can both describe movement into a room or the occurrence of an event. 'kommen' is very general: someone comes to a place, or something happens. 'eintreten' is more precise when someone enters a room, meaning they cross the boundary and go inside. With events, 'eintreten' often sounds somewhat more formal and is especially used for consequences, effects, or expected states. In everyday language, people more often say 'komm rein' (come in) or 'das kann passieren' (that can happen); 'eintreten' is more precise or somewhat more official.

Die Wirkung des Medikaments tritt erst nach einer Stunde ein.
The effect of the medication only takes effect after an hour.
Nach dem Unterricht kommt die Lehrerin kurz ins Klassenzimmer.
After class, the teacher comes into the classroom for a moment.