In Catholicism, A Journey to the Heart of the Faith (Image Books, 2011) Robert Barron explains the nonviolent way of Jesus in the face of evil as a “third” way, apart from violently fighting or fleeing the evil (pp. 48-51):

In words that still take our breath away, Jesus says, “You have heard that it was said, ‘You shall love your neighbor and hate your enemy.’ But I say to you, love your enemies and pray for those who persecute you” (Mt 5:43-44). In order to understand this radical teaching, we have to be clear on what Jesus means by “love” (agape in Matthew’s Greek). Love is not a sentiment or feeling, not merely a tribal loyalty or family devotion. Love is actively willing the good of the other as other. Often we are good or kind or just to others so that they might be good, kind, or just to us in return. But this is indirect egotism, not love. And this is why loving one’s enemies is the surest test of love. If I am good to someone who is sure to repay me, then I might simply be engaging in an act of disguised or implicit self-interest. But if I am generous to someone who is my enemy, who is not the least bit interested in responding to me in kind, then I can be sure that I have truly willed his good and not my own. And this is why Jesus says, “For if you love those who love you, what recompense will you have? … And if you greet your brothers only, what is unusual about that? Do not the pagans do the same?” (Mt 5:46-47). Jesus wants his followers to rise above the imperfect forms of benevolence that obtain among the general run of human beings and to aspire to love the way that God loves: “for he makes his sun rise on the bad and the good, and causes rain to fall on the just and the unjust” (Mt 5:45). […] If we are truly free from our attachments, especially from the attachment to approval, then we can become “sons and daughters” of [this Love], this God and hence conduits of his peculiar grace. […]

The already radical teaching on loving one’s enemies becomes even more intensely focused as Jesus turns his attention to the practice of nonviolence. Giving voice to the common consensus among law-abiding Jews, Jesus declares, “You have heard that it was said, ‘An eye for an eye and a tooth for a tooth.’ 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. If anyone wants to go to law with you over your tunic, hand him your cloak as well” (Mt 5:38-40). It is most important not to overlook the fact that this was, for its time, quite an enlightened, compassionate rule, for many individuals and nations would have felt justified in answering a violent affront with a devastating and disproportionate counterviolence. The seemingly brutal “eye for an eye” rule was in fact an attempt to delimit the retaliatory instinct. But, as we can see, Jesus is uneasy even with this relatively benign recommendation. I fully realize that Jesus’s instruction can sound like simple acquiescence to the power of violence, but we have to probe further. There are two classical responses to evil: fight or flight. When confronted with injustice or violence, we can answer in kind – and sometimes in our sinful world that is all that we can reasonably do. But as every playground bully and every geopolitical aggressor knows, this usually leads to an act of counterviolence, and then still another retaliation until the opponents are locked in an endless round of fighting. Gandhi expressed it this way: “an eye for an eye makes the whole world blind.” The other typical responses to aggression are running away or submitting – and sometimes, given our finite, sinful situation, that is all we can do. But, finally, we all know that ceding to violence tends only to justify the aggressor and encourage even more injustice. And therefore it appears as though, in regard to solving the problem of violence, we are locked in a no-win situation, compelled to oscillate back and forth between two deeply unsatisfactory strategies.

In his instruction on nonviolence Jesus is giving us a way out, and we will grasp this if we attend carefully to the famous example he uses: “To the person who strikes you on one cheek, offer the other one as well” (Lk 6:29). In the society of the time, one would never have used one’s left hand for any form of social interaction, since it was considered unclean. Thus, if someone strikes you on the right cheek, he is hitting you with the back of his hand, and this was the manner in which one would strike a slave or a child or a social inferior. In the face of this kind of violence, Jesus is recommending neither fighting back nor fleeing, but rather standing one’s ground. To turn the other cheek is to prevent him from hitting you the same way again. It is not to run or to acquiesce, but rather to signal to the aggressor that you refuse to accept the set of assumptions that have made his aggression possible. It is to show that you are occupying a different moral space. It is also, consequently, a manner of mirroring back to the violent person the deep injustice of what he is doing. The great promise of this approach is that it might not only stop the violence but also transform the perpetrator of it.

Robert Barron then gives some examples of this strategy, as it was used by Mother Teresa, Desmond Tutu and Pope John Paul II (see the video fragment below):

The award winning social skills educator Brooks Gibbs, who tries to educate people to live by the Golden Rule, took the “mirroring strategy” of Jesus as a way to deal with bullying. What the mirroring strategy does, basically, is to take away the power of the bully to humiliate the victim. The words of the bully are received as “badges of honor”. It is comparable to the way the word “nigga” is sometimes used nowadays among African-American youngsters themselves, so the word gradually loses its power to have a derogatory meaning. I remember being called “homo” one time and thanking the one who yelled it for “the compliment”. In other words, the mirroring strategy in the face of evil is a kind of transformative mimetic practice that invites evildoers to imitate another set of practices and assumptions to end the evil. It could be an example of what René Girard calls “good mimesis”.

Does it work? Not always, but if it does, the results are stunning, as is shown in the video below by Brooks Gibbs: