But I say to you, offer no resistance to one who is evil. When someone strikes you on (your) right cheek, turn the other one to him as well. (Matt 5:39)There are two typical reactions to violence. Fight or flight. Reacting with violence simply escalates the violence. The tougher person wins. Flight confirms the aggressor in his actions and allows the aggression to continue.
Jesus espouses a third way. In those times, you did not use your left hand for most uses since it was typically considered unclean. A person would strike with his right hand. If he strikes your right cheek, he would be using the back of his hand, like you would strike a bad animal. It would show a contemptuous, you're-not-worth-it attitude.
Turning the other (the left) cheek does two things, it forces him to not use the back of his hand, making him unable to strike with the same dismissive attitude and it shows your defiance to his violence in that you are not running away. You mirror his violence back to him as if to say, "I am not cowed."
He gave a wonderful example. Mother Theresa on the streets of Calcutta held a starving child by the hand and took the child to a bakery. She approached the baker and begged some bread for the child. The baker contemptuously spat full in her face. She stood firm, looked at the baker and said "That was for me, now can you give something for the child?"
The baker would have been taken aback by the woman who did not fight back nor flee, but stood firm, showing him her face, asking gently again for bread.
Turning the other cheek is an oftentimes confusing phrase that does not mean pacifism, but resistance to evil without violence and without abandonment. It requires facing the enemy, come what may and showing him that you are not beaten, nor are you even harmed. Nothing he can do can touch your resolve as if to say "the gates of hell shall not prevail against it."