How is Shakespeare's Twelfth Night a romantic play based on a love triangle?
A love triangle is a type of relationship in which three people are involved, forming the three parts of a triangle. While typically in love triangles two people love the exact same person, the term can also apply to three people who are romantically linked in some other way. Twelfth Night actually involves a couple of different love triangles.
The first triangle involves Duke Orsino, Olivia, and Viola. In this triangle Orsino is in love with Olivia who rejects him. Meanwhile, Viola falls in love with Orsino who, thinking that Viola is a man named Cesario, sends Viola to court Olivia for him. As a consequence, Olivia falls in love with Cesario, who is actually Viola, and Viola rejects her. While this triangle does not involve two people who are in love with the exact same person, it does involve three people who are romantically connected, and therefore counts as a love triangle.
The second love triangle does involve two people in love with the same person. This triangle involves Orsino, Olivia, and Sebastian, who Olivia thinks is Cesario. While Orsino is still pining for Olivia, she encounters Sebastian, who she thinks is Cesario, and asks him to come into the house with her so that she can tell him all about Sir Toby's pranks, who had just challenged Sebastian as Cesario to a duel. When Sebastian readily consents, unlike Cesario who would refuse, Olivia very quickly proposes marriage to him, even bringing a priest, as we see in her lines:
Blame not this haste of mine. If you mean well,
Now go with me and with this holy man
Into the chantry by: there before him
And underneath that consecrated roof,
Plight me the full assurance of your faith. (IV.iii.22-26)
Sebastian, being a man, falls in love with Olivia in return and readily agrees to the marriage, leaving Orsino still being the rejected man. Hence, since two men fall in love with the same woman, we see that this is a perfect example of the classic love triangle.