How does Oedipus discover that he killed King Laius?

In Sophocles' play Oedipus Rex, Oedipus discovers he killed King Laius through an obsessive investigation into his own past.1

The process unfolds as follows:

• Initial Accusation: When Oedipus seeks to identify the murderer of Laius to end a plague in Thebes, he summons the blind prophet Tiresias. Though initially reluctant, Tiresias eventually reveals that Oedipus himself is the killer.2
• Investigation: Oedipus conducts a series of tense inquiries, during which he begins to suspect that an old man he killed at a crossroads years earlier was, in fact, King Laius.1
• The Truth Revealed: Through his investigation, Oedipus learns that he was abandoned as an infant by Laius and Jocasta due to a prophecy that their son would kill his father. He discovers that he survived, was adopted in Corinth, and ultimately fulfilled the prophecy by killing his true father and marrying his mother.13

Upon realizing the full extent of his actions, Jocasta dies by suicide, and the guilt-stricken Oedipus blinds himself.14