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Eddard has figured out the big secret (which we knew all the way back in Chapter 9): that Cersei's kids are Jaime's, not Robert's. The big book of lineages showed him that all Baratheon-Lannister marriages result in black-haired kids. That's what Jon Arryn meant by "the seed is strong."
So Eddard plans to ship his kids back to safety and meets Cersei in the godswood (where no one ever goes since they don't much care about the old gods down in the south).
Eddard and Cersei have a civil conversation about incest and murder, and Eddard tells her that she should get out of town before Robert comes back, because he's going to be mad with a capital MAD.
Cersei seems unimpressed (although she does try to seduce Eddard) and lays down the great line: "When you play the game of thrones, you win or you die. There is no middle ground" (80). Booya.