die Annäherung
approach
The movement in which the distance to a person, place, or object decreases.
Die Annäherung des Fuchses an den Hühnerstall blieb unbemerkt.
The fox's approach to the henhouse went unnoticed.

approach / rapprochement / approximation
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approach
The movement in which the distance to a person, place, or object decreases.
Die Annäherung des Fuchses an den Hühnerstall blieb unbemerkt.
The fox's approach to the henhouse went unnoticed.

'Annäherung' (approach/rapprochement/approximation) is a noun derived from the separable verb 'sich annähern' (to approach). The ending '-ung' turns the verb into a feminine noun.
Die Kinder beobachten die Annäherung der Katze an den Vogel.
The children watch the cat's approach toward the bird.
The noun 'Annäherung' is almost always used with the preposition 'an' followed by the accusative.
Die Annäherung an das schwierige Thema erfordert viel Sensibilität.
The approach to the difficult topic requires a lot of sensitivity.
The context shows the meaning: If it is about places or living beings, spatial movement is meant. If it is about opinions, countries, or parties to a conflict, the figurative meaning of reaching understanding is meant.
Die Annäherung des Schiffes an die Küste war schwierig.
The ship's approach to the coast was difficult.
In political or social contexts, people often speak of a 'schrittweise Annäherung' (gradual rapprochement) or of 'eine Annäherung erzielen' (to achieve a rapprochement).
Die Diplomaten arbeiten an einer Annäherung der beiden Länder.
The diplomats are working on a rapprochement between the two countries.
The meaning as 'Approximation' (approximation) is used almost exclusively in scientific or mathematical texts.
In der Physik ist die Verwendung von Annäherungen oft notwendig.
In physics, the use of approximations is often necessary.
distance / removal
'die Entfernung' and 'die Annäherung' are opposites here. 'die Entfernung' describes the distance between two places, people, or things. 'die Annäherung' describes a movement in which this distance becomes smaller. So you use 'Entfernung' when you want to know how far away something is. You use 'Annäherung' when something or someone is coming closer. The contrast is: with 'Entfernung', distance exists or is thought of as large; with 'Annäherung', the distance becomes smaller.

Die langsame Annäherung des Zuges ist am Bahnsteig gut zu sehen.
The train's slow approach can be clearly seen from the platform.
Auf dem Amt wurde heute die Entfernung zwischen den beiden Grundstücken geprüft.
At the government office today, the distance between the two properties was checked.