What is a Tragic Hero?

Looking over my notes in preparation for the midterm in English, I recall discussing how both Oedipus and Medea do not completely conform to the ideals of Artistotle's tragic hero. For example, Medea cannot truly identify as a tragic protagonist or hero because she never gets a justified punishment, rather, ends with everything turning out how she wanted them to. Medea ends up more as the antagonist in the sense that whereas the reader once pities her in the beginning of the play, we ultimately realize her evil, manipulative ways which changes our viewpoint of her character. In Oedipus, the debate over free will vs. fate is discussed. If Oedipus's actions are predestined, then he can't be classified as a tragic protagonist because his downfall did not stem from his own error in judgment, rather, something he couldn't control. This signifies him as the antithesis of the typical tragic hero. These two plays do not completely conform to Aristotle's definition of a tragic protagonist.