As soon as Creon returns from the Oracle at Delphi, Oedipus...
Oedipus vows to the people of Thebes that he will find Laius's murderer and save them from the famine and plague they're enduring. He's impatient to solve the mystery and prove that he deserves the title of "world-renowned king."
As soon as Creon returns from the Oracle at Delphi, Oedipus peppers him with questions about Laius's murder. All he learns is that the original investigation was incomplete due to the appearance of the Sphinx at Thebes.
OEDIPUS. What trouble can have hindered a full quest,
When royalty had fallen thus miserably?
CREON. The riddling Sphinx compelled us to let slide
The dim past and attend to instant needs.
Oedipus is undeterred, however, and uses the opportunity to take charge of the investigation.
OEDIPUS. Well, I will start afresh and once again
Make dark things clear.
Oedipus has already sent for Teiresias, the seer-prophet. He is getting annoyed that Teiresias hasn't yet arrived to provide his insight about the matter.
CHORUS. My liege, if any man sees eye to eye
With our lord Phoebus, 'tis our prophet, lord
Teiresias; he of all men best might guide
A searcher of this matter to the light.
OEDIPUS. Here too my zeal has nothing lagged, for twice
At Creon's instance have I sent to fetch him,
And long I marvel why he is not here.
Teiresias arrives and almost immediately asks to go back home. Oedipus insists that Teiresias stay and orders him to tell what he knows, but Teiresias refuses to speak about the matter.
OEDIPUS. What then, thou knowest, and yet willst not speak!
Wouldst thou betray us and destroy the State?
Oedipus calls Teiresias a "monster" and rages at him, but Teiresias isn't intimidated by Oedipus.
TEIRESIAS. I have no more to say; storm as thou willst,
And give the rein to all thy pent-up rage.
Oedipus has already demonstrated his impatience in his dialogue with Creon. Now even more impatient and angry, Oedipus accuses Teiresias of plotting to kill Laius and attempting to cover up his deed with silence.
OEDIPUS. Yea, I am wroth, and will not stint my words,
But speak my whole mind. Thou methinks thou art he,
Who planned the crime, aye, and performed it too,
All save the assassination; and if thou
Hadst not been blind, I had been sworn to boot
That thou alone didst do the bloody deed.
Oedipus pushes Teiresias too far. In response, Teiresias reveals what he knows, to Oedipus's own detriment.
TEIRESIAS. ... Thou art the man,
Thou the accursed polluter of this land...
I say thou art the murderer of the man
Whose murderer thou pursuest.
Oedipus is enraged even further, he and accuses Teiresias of conspiring with Creon to disgrace him and to use this opportunity to usurp his throne.
The Chorus attempts to intervene and impart some reason to the heated discussion between Oedipus and Teiresias. They keep arguing and angrily insulting each other until Teiresias has had enough of Oedipus's browbeating and threats. Teiresias tells Oedipus that not only did Oedipus kill Laius, but that Laius was his father. Therefore, Oedipus married his own mother, Jocasta, fulfilling his fate and the prophecy of the gods—the fate they made every effort to avoid—that Oedipus would kill his father and marry his mother.
As the play goes on, Oedipus shows himself to be equally impatient and quick to anger with Creon—to whom he gives a choice of exile or death for conspiring against him—and with the Herdsman, who, like Teiresias, is reluctant to speak.
Oedipus threatens the Herdsman with torture and death if he doesn't tell what he knows. Like Teiresias, the Herdsman tells a story that leads to Oedipus's tragic fall from king to murderer, and then to exile.