- The ThreeAmigos are GradyBooch, IvarJacobson, and JimRumbaugh.  These three got together a few years ago to unify the different processes and notations that were factionalizing the object-oriented world.  IvarJacobson and JimRumbaugh joined GradyBooch at Rational and the result so far has been the UnifiedModelingLanguage which is a fairly well received ObjectModelingNotation for modeling object-oriented systems. 

The ThreeAmigos term itself is ostensibly used in order to ''not'' have to remember the names or look up the spelling of the names of the ThreeAmigos (see, I just did myself).
-- DionHinchcliffe
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It also plays on the fact that the three of them used to be quite frosty to each other until the lure of the $$$ persuaded them to take a common position.  (MissionaryPosition?)  Now of course they're the best of pals...

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It's also the name of a rather silly (but funny) movie starring Steve Martin, Chevy Chase, and Martin Short. 

Not to be confused with the Three Caballeros, a disney cartoon with DonaldDuck, Joe Carioca (a Parrot from Brasil) and Panchito (a Mexican Rooster).

The term is an example of the ReuseGroupName pattern. 
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The Tree Amigos is the name of a Agile pattern whereby a Product Owner, a Developer and a Quality Assurance person come together to flesh out a story on starting work. Sometimes the phrase is also used for when the three people come together at the completion of a story to confirm that it is running, tested and fulfils all the acceptance criteria. I'd like to know where the use of the phrase for this purpose originates, if anyone knows.