How does the quote below from Romeo and Juliet demonstrate that hate can be caused by love and be overcome by it?
”Here’s much to do with hate, but more with love.
Why then, O brawling love, O loving hate,
O anything, of nothing first created.”(I.i.169)
This is a very famous quote from this excellent tragedy, and, as you have identified, has a lot to do with the relationship between love and hate as expressed in this play. This quote uses a series of paradoxes to point out the idiocy of the situation that plagues the households of the Capulets and the Montagues.
Let us remind ourselves of the context of this scene. Benvolio has drawn near to Romeo to talk to him about his aloof behaviour and how depressed he appears to be, and Romeo, trying to change to subject, remarks on the "fray" that he obviously sees some form of evidence of and utters the quote that you have cited. Note how he refers to the violence as having much to do with "hate," but actually being evidence more of "love"--the love and pride that each family has in its own identity. It is for love of their household that the members of the two households feud, not hatred, which thus leads to the two paradoxes that Romeo uses: "O brawling love, O loving hate." As for the second part of your question, I don't necessarily think that these lines alone show that hate can be overcome by love. Rather I think they just explain the paradox that lies at the heart of the violence of the play. To find evidence of how hatred is overcome by love, look at how the deaths of Romeo and Juliet bring peace to the feud.