Peter (‘Cephas’ in Greek) was inconsistent.  He had stopped living according to Jewish customs because he rightly believed that God accepted him because he trusted in Jesus alone for his salvation (Galatians 2:12).   But he was teaching Gentiles (non-Jews) to be circumcised and obey the Jewish food laws.  He was wrong.  That was not just a private error in his personal life; it affected many other people and got in the way of the simplicity of the gospel.
 
Private sin should be dealt with personally and out of the public gaze; but when error infects other people, a public rebuke is necessary (Matthew 18:15-17).  Was Paul brave or foolhardy in challenging Peter in that way?  Neither!  Paul was simply being faithful to the truth of the gospel and pointed out Peter’s inconsistency to cause others to see that Peter had not set the right example.  Those who followed him were foolish too.
 
Often the more thought-provoking rebukes come in the form of a question.  Jesus used that method of teaching, and a good example is in Luke 14:1-6.  It is also a wonderfully direct tool to encourage our friends and colleagues to consider the inconsistencies in their lifestyles, especially if they claim to follow Jesus.  Truly wise people will challenge others to think by asking them questions.

When the integrity of the gospel is at stake, it is important to help people to understand the implications of their behaviour so that they can repent and be restored.  So instead of arguing with people and telling them what you think, why not ask them about the consistency of their lives?  It is a fair question even in public and will help others to think as well.