Why does Hamlet have that specific play presented at court?
Hamlet has that specific play presented at court because it contains a scene of the murder of an old king killed in the exact manner in which Claudius killed Hamlet's father. In doing this, he hopes to provoke a response from Claudius that will reveal his guilt.
Why, in act 3, scene 2 of Hamlet, does the bereft prince choose the specific play that he does to be performed for members of the royal court? Because he hopes its similarities to the circumstances of his own father's death will provoke a response from his killer, Claudius, and thus prove his guilt.
In act 1, Hamlet learns from the troubled spirit of his deceased father, King Hamlet, that he was murdered in his sleep by his brother Claudius so that the latter might seize the crown and the queen. Yet, before acting on this knowledge, Hamlet feels the need for a more substantial proof of Claudius' guilt than the words of a wrathful wraith, however seemingly familiar.
Thus, inspired by the arrival of a troupe of traveling players, Hamlet commands them to enact some scenes from a play, The Murder of Gonzago, which features the murder of an old king by his brother. For good measure, the prince also inserts a few additional lines which he hopes will trigger a guilty reaction from Claudius. He amuses himself by re-titling his altered work "The Mousetrap."
At the crucial point in the performance of the play, the actor/murderer pours poison into the ear of his sleeping victim in imitation of the manner in which Claudius had killed King Hamlet. As his son narrates in verse,
He poisons him i' th' garden for ’s estate. His name’s Gonzago. The story is extant, and writ in choice Italian. You shall see anon how the murderer gets the love of Gonzago’s wife.
In shock, Claudius quickly rises and runs off with the rest of the court soon to follow. Hamlet now has confirmation of the truth of the Ghost's tale:
HAMLET: O good Horatio, I'll take the ghost's word for a thousand pound. Didst perceive?
HORATIO: Very well, my lord.
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