"Anne is ON the point of leaving" sounds incorrect to me. I don't know if it's because of my local region, but I would quite definitely say "Anne is AT the point of leaving."
Duolingo fails to accept the two better English alternatives, Anne is on the verge of leaving and Anne is on the point of leaving.
sur le point de is a French prepositional phrase meaning "about to". I agree that "on the verge of" and "on the point of" are also valid translations, as does Larousse:
Did you report them?
The connotations of these different compositions may be different, still--but who can explain them?
but 'Anne is at the point of leaving' is not accepted. I think it ought to be. I pondered the correct translation shown above but thought a somewhat more literal translation would be a safer bet.