Michael Strahan and Nicole Murphy -- Breakup Announcement Triggered by 'Proof' of Other Woman

Nicole Murphy decided to tell the world she had broken up with Michael Strahan -- right when he was entering the Hall of Fame -- because that's when she "confirmed" he was unfaithful.

A source close to the couple tells TMZ ... Nicole first got wind Michael might be dating another woman in early July and started investigating ... the way scorned women do. We're told she was finally convinced it was true on Friday and immediately decided to go public.

The source did not know the nature of Nicole's proof.

We're told Nicole insists her timing wasn't out of revenge, rather a total coincidence because that's when she knew for sure. It just happened to come precisely when Michael was getting the greatest honor of his career.

Other sources tell us Nicole and Michael have been broken up for several weeks -- and, in fact, their bi-coastal relationship has been on and off for almost 2 years ... since Michael took the 'Live' gig.

Ultimately this may come down to that old "Friends" theory -- but we're told Nicole definitely did NOT think they were "on a break."

Reps for Michael and Nicole have not responded to calls for comment.