pivot / crux
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crux
The central and decisive point of a matter, around which everything revolves. It is the core of a problem or discussion.
Der Angelpunkt der Debatte ist die Finanzierung des Projekts.
The crux of the debate is the financing of the project.
The word is made up of 'die Angel' (hinge) and 'der Punkt' (point). An 'Angel' is a hinge, for example on a door, so it is the point around which something turns. This literal meaning also explains the figurative sense.
Der Angelpunkt der ganzen Geschichte ist ein Missverständnis.
The crux of the whole story is a misunderstanding.
You can recognize the figurative meaning from abstract nouns like 'Diskussion' (discussion), 'Problem' (problem), or 'Frage' (question). You can recognize the literal, technical meaning from concrete objects like 'Tür' (door), 'Hebel' (lever), or 'Konstruktion' (construction).
Der Angelpunkt des Problems ist der Mangel an Kommunikation.
The crux of the problem is the lack of communication.
In the figurative meaning, the word is often used to describe the core of a matter. Common combinations are 'der Angelpunkt der Diskussion' (the crux of the discussion), 'der Angelpunkt der Frage' (the crux of the question), or 'der springende Punkt' (the crucial point).
Datenschutz ist der Angelpunkt bei der Einführung der neuen Software.
Data protection is the crux when introducing the new software.
core / pit / nucleus
'der Kern' and 'der Angelpunkt' can both mean the central point of something. 'der Kern' describes more the essential part or the inner content of a problem, thought, or topic. 'der Angelpunkt' emphasizes more strongly that everything depends on this point or revolves around it. The word fits especially well when a point is decisive for a decision, solution, or development. In short: 'Kern' asks about what is essential; 'Angelpunkt' asks about the decisive pivot point.
Beim Streit mit den Nachbarn war der Lärm am Abend der Angelpunkt.
In the dispute with the neighbors, the noise in the evening was the crux.
Der Kern der Aufgabe ist einfacher, als er zuerst wirkt.
The core of the task is simpler than it first seems.