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Suppose I have a picture of something. Suppose further that there is a cut in the middle of the picture.

So, is it correct to say:

There is a cut in the middle of the picture

or

There is a cut at the middle of the picture

and Why?

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It's correct to say in the middle as this is idiomatic and comes naturally to native English speakers. Google Books Ngram Viewer indicates a strong return for in the middle and virtually none for at the middle. If you ask why, it's because that's the way the language has developed.

In the examples you give, the word middle is a noun. However, you can also use it as an adjective, as in the middle point or the middle line. Then the preposition you would use is likely to change.

Most people would say at the middle point (because we talk about at a point) and on the middle line (because we say things are on a line or sometimes in a line, which means something different.)

These are conventions rather than rules and are not absolute. Depending on the context, other prepositions may be suitable, e.g.: in the line of fire and on the point of political correctness.

https://books.google.com/ngrams/graph?content=at+the+middle%2Cin+the+middle&year_start=1800&year_end=2000&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Cat%20the%20middle%3B%2Cc0%3B.t1%3B%2Cin%20the%20middle%3B%2Cc0

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