How would one define the relationships in Shakespeare's Twelfth Night?
Many different relationships can be found in Twelfth Night. Some of those relationships include friendships and false friendships, and there is even an antagonizer-victim relationship that gets flipped around.
Maria proves to have somewhat of a friendship with Sir Toby, even though she shows she greatly disapproves of his behavior. We especially see their friendship in their interactions in Act 1, Scene 2 as well as other scenes in which they plan and execute the prank against Malvolio. Even though Maria scolds him, warning that Olivia is very offended by his behavior, their friendly banter shows their friendship. Act 1, Scene 2 also shows Sir Toby's friendship with Sir Andrew, which proves to be a false friendship. We already see it being implied that Toby has only brought Andrew into the house to court Olivia because Toby wants to take advantage of Andrew's money, which we see when Maria asks why he brought such a fool as Sir Andrew into the house and one of Toby's replies is, "Why, he has three thousand ducats a year," showing us just how interested Toby is in Andrew's money (I.iii.20). Later, Toby's false friendliness to Andrew is exposed when Toby declares what he really thinks of Andrew, "Will you help an ass-head, and a coxcomb, and a knave? a thin-faced knave, a gull?" (V.i.212-13).
The antagonizer-victim relationship is first shown in Malvolio's antagonism of the other characters, such as insulting Feste's abilities as a court jester and getting on the other characters' nerves for being such a self-righteous "Puritan" (II.iii.129) and telling Sir Toby he should leave the house if he wishes to continue his disorderly drinking. But the other characters turn the tables on this relationship and become the antagonizers while Malvolio becomes their victim through their prank.
One type of relationship found in Twelfth Night is the love triangle. There are actually two different love triangles found in Twelfth Night. Typically, a love triangle usually involves two people who are in love with the same person; however, there can be variations on the theme, just so long as all three people have some sort of relationship amongst each other.
The first love triangle involves Duke Orsino, Olivia, and Cesario. Duke Orsino believes he is in love with Olivia, but Olivia falls in love with Viola, who is pretending to be Cesario. Meanwhile, Viola also falls in love with Orsino, which completes the triangle. The unrequited love in this triangle functions as the main conflict in the play. The resolution occurs when Olivia mistakes Sebastian for Cesario and marries him instead; the resolution further occurs when Orsino learns that Cesario is truly a woman named Viola and decides to marry her instead. We especially see the resolution developed in the final scene. Sebastian remarks on the advantage of Olivia having married Sebastian rather than the real Cesario, pointing out that had she married Cesario, Olivia "would have been contracted to a maid" (V.i.271). Furthermore, as soon as Orsino learns Cesario's real identity, he makes the decision to join in on the happy moment by wanting Viola for his wife instead of Olivia, as we see in his lines, "If this be so, as yet the glass seems true, / I shall have a share in this most happy wreck" (275-76).
The fact that Olivia marries Sebastian instead brings us to the second love triangle in the play. Olivia is in love with Cesario, who rejects her because Cesario is actually a woman. Meanwhile, Olivia meets Sebastian, who falls in love with Olivia immediately, forming the third corner of the love triangle.