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.