I came across this:

The phone rang when Tom was on the point of leaving.

As I usually use "at the point of leaving", are both possible?

  • Use "at". When I read "on", I think "Tom was (on the brink of) leaving." – user3169 Jan 13 '19 at 1:59

I would use on, personally.

Google Books NGram Viewer suggests that both are possible, but that on is the more common of the two:

the point of leaving

If I look at some of the actual examples of the at phrasing gathered by Google, it seems that it's sometimes used in a more literal sense (the banks of the bayou at the point of leaving the river), even though I see examples of its figurative use too.

So, both can be used. It's up to you to decide which you prefer.

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