Definition: Demophon was one of Theseus' legal heirs, along with his (half-)brother Akamas (Acamas). In lists of kings of Athens during the legendary period of Greece, Demophon comes after Menestheus who follows Theseus. Menestheus was not a son of Theseus, but a descendant, possibly like Theseus, of Poseidon. Demophon's mother may have been Ariadne, in which case, Akamas was his full brother. Pindar names Antiope as Demophon's mother. Apollodorus names Demophon and Akamas as children of Theseus and Phaedra. Antiope is another possible mother of the brothers.
In the epic cycle, but not the Iliad, Demophon and Akamas join the Trojan War forces in order to take the opportunity to rescue their grandmother Aethra. They succeed after the fall of Troy. Demophon returns to Athens to become king after the childless Menestheus dies.
Demophon grants the children of Hercules refuge in Athens in Euripides' Heracleidae.
See Who Was King of Athens During the Trojan War?
Source: Timothy Gantz: Greek Myth
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It is supposed to have been during the reign of Demophon that Athena acquitted Orestes at his matricide trial (Fragmenta Historicorum Graecorum, ed. C. Müller, i.546). There is another mythological Demophon -- of Eleusis.

