sich nähern
to approach
To reduce the physical distance to someone or something; to come closer.
Langsam näherte sich der Zug dem Bahnhof.
Slowly, the train approached the station.

to approach
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to approach
To reduce the physical distance to someone or something; to come closer.
Langsam näherte sich der Zug dem Bahnhof.
Slowly, the train approached the station.

The verb 'nähern' (to approach) is almost always used reflexively: 'sich nähern' (to approach). The object that one approaches is always in the dative. This is a fixed rule for this verb.
Der Wanderer näherte sich langsam dem Gipfel des Berges.
The hiker slowly approached the summit of the mountain.
The verb 'nähern' (to approach) is derived from the adjective 'nah' (near) or its comparative form 'näher' (nearer/closer). It literally means to reduce the distance to something or to come 'näher' (closer) to something.
Pass auf, ein Auto nähert sich von links.
Watch out, a car is approaching from the left.
The context shows the meaning: If it is about a place or a person, physical approach is meant. If it is about a topic or problem, mental engagement is meant. If it is about numbers or values, becoming more similar or closer is meant.
Das Schiff nähert sich dem Hafen.
The ship is approaching the harbor.
In the abstract meaning, 'sich nähern' (to approach) is often connected with intellectual or conceptual goals: 'sich einem Thema nähern' (to approach a topic), 'sich einer Frage nähern' (to approach a question), 'sich einer Lösung nähern' (to approach a solution).
In ihrer Forschung nähert sie sich dem Phänomen aus historischer Perspektive.
In her research, she approaches the phenomenon from a historical perspective.
to resemble
'sich nähern' and 'ähneln' can both have to do with similarity. But the difference is important: 'ähneln' describes a state. Two people or things are already similar. 'sich nähern' describes a development. Something is becoming more and more similar to a value, state, or quality. Learners may confuse the words because both often appear in comparisons. If the similarity is already there, 'ähneln' fits. If it is only developing or becoming stronger, 'sich nähern' fits.

Mit jedem Training nähert sich seine Leistung dem Niveau der Profis.
With each training session, his performance is approaching the level of the professionals.
Meine Freundin ähnelt ihrer Schwester im Aussehen und im Humor.
My friend resembles her sister in appearance and in humor.