Dear all,
I have seen the "Are you finished eating? " in my TOEIC book.

But I think "Do you finished eating? or Have you finished eating? " is more understandable for me than that.
Would you please give me an idea why "Are you finished eating?" is correct.
Thanks in advance.
"Are you finished eating" is correct "Have you finished eating" is also correct and they are mostly interchangeable.
1. (to) finish = present tense
2. (had) finished = past tense
3. (am) finishing = present participle

"Are" works with:

1 Are you to finish (doing something in the present progressive tense)?(inquiry about something to take place in the future)
2. Are you finished (doing something in the present progressive tense)?(inquiry about something taking place now or in the past (imperfect tense, I believe))
3. Are you finishing (doing something in the present progressive tense)?(inquiry about something taking place now or in the future)
1. (to) have = present tense
2. (have) had= past tense
3. (am) having = present participle

"Have" works with:

1 Have you to finish (doing something in the present progressive tense)?(inquiry about something to take place in the future)

2a. Had you to finish (doing something in the present progressive tense)? (inquiry about something taking place and completed in the past (perfect past tense, I believe))
2. Have you finished (doing something in the present progressive tense)?(inquiry about something taking place now or in the past (imperfect past tense, I believe))
2a. Had you finished (doing something in the present progressive tense)? (inquiry about something taking place and completed in the past (perfect past tense, I believe))
"Do you finished eating" is not correct.
1. do (does = third person singular) = these are both some type of presenttense

2. did = past tense
3. doing = present participle

"Do" works with:

1 Do you finish (doing something in the present progressive tense)?(this is a general inquiry about something to take place possibly more than once or repeatedly)
2. Did you finish (doing something in the present progressive tense)?(inquiry about something taking place now or in the past (perfect past tense, I believe))
To say the finishing is occurring now, one must use "is." Example: "He is finishing eating now." or "Are you finishing eating now?"

Maybe someone else can provide more detail than my answer.

I hope this helps.
Regards,
Dan
Dear all, I have seen the "Are you finished eating? " in my TOEIC book.

Characteristic American English usage.
But I think "Do you finished eating?

No, no, nopity no.
or Have you finished eating? " is more understandable for me than that.

Fine, Present perfect - the plate is still in front of the person eating, standard in both BrE and Ame.
Would you please give me an idea why "Are you finished eating?" is correct.

because your book is for TOEIC (US) not IELTS (Rest of the English-speaking world).
Thanks in advance.

You're alright.
DC
Students: Are you brave enough to let our tutors analyse your pronunciation?
"Do you finished eating? Incorrect
Did you finish eating? Correct