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Achilles was part of the Achaean force Agamemnon took to Troy to win back Helen for Agamemnon's brother Menelaus. Proud and autocratic Agamemnon antagonized Achilles, and so Achilles sat out most of the fighting. Achilles was at length motivated by revenge to join the fray when his friend Patroclus is killed by Hector, the greatest of the Trojans. In a rage, Achilles killed Hector, and then dishonored the body by dragging it around in a chariot for nine days, until the father of Hector, King Priam, appealed to the better nature of Achilles and prevailed upon him to return the corpse to his family in Troy for proper funeral rites.
Achilles was the son of the mortal Peleus and the nymph Thetis. Thetis tried to make her son immortal by dipping him into the River Styx, while holding him by his ankle.
His ankle was therefore the only portion of him capable of sustaining a mortal wound, which he received from a goddess-guided arrow shot by Paris of Troy. The mortality of Achilles is also explained as having been caused by an unsuccessful application of the treatment for immortality -- ambrosia by day and fire by night, which was a technique the goddess Demeter once tried.