to determine / notice / secure
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to notice
To notice, recognize, or come to a particular realization about something.
Als ich zu Hause ankam, musste ich feststellen, dass ich meinen Schlüssel vergessen hatte.
When I arrived home, I had to notice that I had forgotten my key.
'Feststellen' is a separable verb. The prefix 'fest-' goes at the end in a main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Der Techniker stellt einen Fehler fest.
The technician detects an error.
The verb is made up of 'fest' (stable, fixed) and 'stellen' (to place). The literal meaning is 'to fix something in place' (meaning 3). From this comes the figurative meaning of mentally 'fixing' a fact or realization (meanings 1 and 2).
Bitte stellen Sie die Räder des Kinderwagens fest, damit er nicht wegrollt.
Please lock the wheels of the stroller so it doesn't roll away.
The context reveals the meaning: If it is about a mechanical part (brake, wheel), 'fixieren' (to fix/secure) is meant. If a 'dass' clause follows, it usually means 'bemerken' (to notice). If official people are speaking (doctor, police), it often means 'offiziell konstatieren' (to officially establish/state).
Der Arzt stellte eine deutliche Besserung fest.
The doctor noted a clear improvement.
In the meaning of 'bemerken' (to notice) or 'erkennen' (to recognize), 'feststellen' is very often used with a subordinate clause that begins with 'dass'. This is used to report one's own observation or realization.
Er musste feststellen, dass sein Plan nicht funktionierte.
He had to notice that his plan wasn't working.
In a formal or official context, 'feststellen' is often used to establish facts. Typical collocations are 'den Tod feststellen' (to pronounce/confirm death), 'die Identität feststellen' (to establish identity), or 'einen Schaden feststellen' (to determine damage).
Die Polizei konnte die Ursache des Unfalls schnell feststellen.
The police were able to quickly determine the cause of the accident.
to recognize / realize
'erkennen' and 'feststellen' can both mean that you come to a realization. 'erkennen' puts more emphasis on the moment of understanding or perceiving: you grasp something or see what is going on. 'feststellen' sounds more factual and often more precise: you check something and arrive at a result. In official or medical situations, 'feststellen' is especially natural. The words can be confused because both can fit with problems, mistakes, or facts. 'erkennen' fits more with inner understanding, while 'feststellen' fits more with a clear result.
Die Ärztin stellte fest, dass der Patient Fieber hatte.
The doctor determined that the patient had a fever.
Nach der Nachricht erkennt sie, dass sie den Termin vergessen hat.
After the message, she realizes that she has forgotten the appointment.