The first time Hector meets Achilles on the battlefield, in Book IX of the Iliad, he is prudent enough to avoid dueling the Myrmidon superhero. However, in Book XXII Hector decides to face Achilles in single combat, a decision that costs him his life.
Why does Hector accept Achilles' challenge?
Pride and shame
It is put best in book 12, 90-130:
BkXXII:1-89 Priam and Hecabe fail to dissuade Hector | poetryintranslation.com
Which shows his own thinking on the issue: [bolding mine]
But his [Hector's] proud thoughts were troubled: ‘Alas, if I retreat through the gate, to the safety of the wall, Polydamas will not be slow to reproach me, since he advised me to withdraw our forces to the city, on that fatal night when Achilles re-appeared. I refused, though it may have been better! Now, in my folly, having brought us to the brink of ruin, I’d be ashamed to hear some insignificant Trojan, or his long-robed wife, say: ‘Hector has brought ruin on the army, trusting too much in his own right arm.’ If that’s what they’ll say, then I’d be better by far to meet Achilles face to face and kill him before returning to the city, or die gloriously beneath its walls. Of course, I could ditch the bossed shield and heavy helmet, lean my spear on the wall, and go and promise peerless Achilles to return Helen and her treasure to the Atreidae, all that Paris brought in the hollow ships to Troy, to begin this strife. I could say too that we’ll then divide all the remaining treasure in the city, and then induce the Elders to state on oath that they’ll conceal no part of that treasure, but grant half of all the lovely city holds. But what’s the point of such thoughts? I’ll not approach him like a suppliant only to have him show neither mercy nor respect, but kill me out of hand, stripped of my armour and defenceless as a woman. This is no lover’s tryst of lad and lass, by oak or rock! Lad and lass, indeed! Better to meet in bloody combat, now, and see to whom Zeus grants the glory!
