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**He rose when Ellen appeared, and asked how she did, and offered her a chair. She thought that he had never looked better.**

I think that in the above sentence, 'he asked how she did' is the reported speech and the direct speech is ' he said to her, 'How do you do?'.

I think that she is acquainted with him because of the sentence ' he had never looked better.

If so, I wonder why he greeted with 'how do you do'. Because I think it's a greeting used when you first meet someone you don't know.

I think 'how are you?' is better greeting than 'how do you do?'. What do you think of that?

I want to know why 'how do you do' was used in the above sentence.
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"Asked how she did" is unheard of. The friendly version is "How are you doing?"
The reported speech version might be, "He asked her how she was doing." I'd say this is even more common than to to say, "He asked her how she was," although "How are you" is somewhat more formal than "how are you doing," while clearly less formal than "How do you do."

Your story may have more context explaining why the more formal greeting was used with an old acquaintance.

This may not be an example of contemporary style. As I say, I've never heard "he asked how she did."
There are only 500 Google hits, and they're all "asked how she did it."
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It would be useful to know the date of the excerpt. English changes, especially between centuries. I would think given the final phrase (he had never looked better" that the two characters already knew each other and therefore "How do you do?" would be inappropriate.

But maybe in Victorian England (i.e. 19th century England) such formal address was used for a longer period of time than in present day (today you use it only when you first meet someone).

Here is an example of a change that can be seen in 19th century English. In Jane Austen, we will read such things as "the letter is come" instead of "the letter has come".

Note in passing that is often difficult to be 100% sure of what the direct speech version would be. Indirect speech is therefore highly suspect- but makes for fun textual analysis!
DarcyI think that in the above sentence, 'he asked how she did' is the reported speech and the direct speech is ' he said to her, 'How do you do?'.

I agree. Unless this is old writing and usage has changed in this respect, my guess is that it's an intentionally humorous way of expressing this. As others have mentioned, "asked how she did" is not standard usage, at least not in modern English.