Instead of taking everything away from Jason and preventing the Corinthians from killing her children, why does Medea find it necessary to kill her children?

Medea finds it necessary to kill her children primarily because it will cause the maximum pain to Jason. Also, when their deaths are combined with the recent murder of his wife, it effectively means he cannot have any male descendants. By killing her children, Medea ensures an end to Jason's lineage.

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The terrible acts of the title character of Euripides's Medea would seem to make her one of the most barbaric characters of Greek drama. In fact, she is the only character in the surviving canon that kills kin, her own children, in cold blood rather than in a moment of derangement.

The profound misery of Medea and the hatred she feels toward the husband who has left her for a royal princess who will bear him more children issues forth from her like a river of bile as the play opens. Worse, her harsh criticism of Jason's father, Creon, over this humiliation, has led to an order for her banishment from Corinth.

...this unexpected blow that's hit me
has destroyed my heart. My life is over,
dear friends. I've lost all joy. I want to die.
The person who was everything to me,
my own husband, has turned out to be the worst of men. (257-262).

Aware that in exile she and her children will become friendless vagabonds, Medea succeeds in persuading the King of Athens, Aegeus, to provide them with sanctuary. In the following monologue, though, she reveals her heinous plan: She wants to murder Jason's beloved princess Glauce with a poisoned garment. More horribly, she plans to kill her own children to punish her ex-husband for his betrayal.

By the end of the play Glauce has died horribly, literally consumed by the gift of a golden gown, and Medea has butchered her sons. She triumphantly confronts Jason with their bodies in a winged chariot. The traumatized father pours out a monologue of grief and hatred before Medea speaks:

MEDEA: Do you think an insult to a woman is something insignificant?
JASON: Yes, to a woman with good sense. But to you it is completely evil.
MEDEA: Well, your sons are gone, that should cause your pain. (1630-1634).

Jason's unbearable pain is Medea's reward.

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