einweihen
to inaugurate
To officially put a new building, facility, or monument into use with a formal ceremony.
Nächste Woche wird die neue U-Bahn-Linie feierlich eingeweiht.
Next week, the new subway line will be ceremoniously inaugurated.

to inaugurate / initiate
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to inaugurate
To officially put a new building, facility, or monument into use with a formal ceremony.
Nächste Woche wird die neue U-Bahn-Linie feierlich eingeweiht.
Next week, the new subway line will be ceremoniously inaugurated.

'Einweihen' (to inaugurate / initiate) is a separable verb. The prefix 'ein-' goes at the end in a main clause. In a subordinate clause, the verb stays together, and in the infinitive with 'zu', 'zu' goes in the middle.
Der Bürgermeister weiht die neue Schule ein.
The mayor inaugurates the new school.
The verb comes from 'weihen', which means 'to consecrate' or 'to dedicate' and has a ceremonial character. So 'einweihen' literally means to dedicate something to its purpose through a first, special action.
Lasst uns die neue Kaffeemaschine feierlich einweihen!
Let's ceremoniously christen the new coffee machine!
When informing someone about a secret, the fixed structure 'jemanden (accusative) in etwas (accusative) einweihen' is used. The preposition 'in' is crucial here.
Ich muss dich in meinen geheimen Plan einweihen.
I have to let you in on my secret plan.
In the meaning of 'offiziell eröffnen' (to officially open), 'einweihen' (to inaugurate) is typically used for public structures such as 'ein Gebäude' (a building), 'eine Brücke' (a bridge), 'ein Museum' (a museum), or 'ein Denkmal' (a monument).
Der neue Park wurde am Samstag vom Bürgermeister eingeweiht.
The new park was inaugurated by the mayor on Saturday.
to open
'eröffnen' and 'einweihen' can both be used for new public places or buildings. 'eröffnen' means that something becomes accessible to visitors or customers for the first time. 'einweihen' (to inaugurate) puts more emphasis on the official or ceremonial handover, often with guests, speeches, or a ceremony. A building can be inaugurated and then opened, or both can happen on the same day. The confusion arises because both words fit with new buildings. 'eröffnen' is more about the question: From when can people use it? 'einweihen' is more about the question: Is there an official ceremony or handover?

Der Bürgermeister weiht morgen die neue Sporthalle ein.
The mayor is inaugurating the new sports hall tomorrow.
Das neue Museum wird am Samstag für die Besucher eröffnet.
The new museum will open to visitors on Saturday.