What is the polite way to show your appreciation when someone waits for you to enter an entrance before they do?They usually say "after you"? What I am supposed to answer?
You basically have two choices
Thank you that's very kind
and continue walking in, or saying
No, please, after you
Please, no, after you
stopping before entering, extending your arm to hold the door open, and then holding the door while the person previously holding the door for you passes through.
Protocol dictates that if a person has been holding the door, say for a large party to go through, you would relieve that person of such duty, if you are suitably able, when it becomes your turn to pass. The other person may or may not be travelling in the same direction as you are.
Protocol also dictates that the least burdened holds the door, with the exception of
Chivalrous custom which dictates Ladies First except when going through revolving doors
The correct is to say "thank you" and to go through.
Insisting that a woman, older person, etc., go through before you, when they have expressed a preference that you go first, would be less respectful than honoring their preference.
I once almost got into a deadlock situation in an elevator with a distinguished computer scientist (Turing Award winner) when I was a graduate student. When the elevator door opened, I automatically paused to let my superior exit first, while he, an older gentleman, waited for me to exit, my being a woman. As soon as I realized what was going on, I exited, thinking that more respectful than insisting on my preferred protocol.