A DANCE WITH DRAGONS page 577
"Prince Aerys … as a youth, he was taken with a certain lady of Casterly Rock, a cousin of Tywin Lannister. When she and Tywin wed, (Aerys) drank too much wine at the wedding feast and was heard to say that it was a great pity that the lord’s right to the first night had been abolished… the liberties (that Aerys) took during the bedding." Ser Barristan to Dany
A FEAST FOR CROWS
"The Imp is no longer my brother, if he ever was." Cersei Lannister
"… but Tyrion is Tywin’s son, not you. I said so once to your father’s face, and he would not speak to me for half a year." Aunt Genna to Jaime
A STORM OF SWORDS
"Aerys had chosen [Jaime] to spite his father, to rob Lord Tywin of his heir."
King Aerys was Cersei and Jaime’s father.
Thats where the penchant for incest comes from, thats why Cersei is completly and utterly insane, thats why she enjoyed burning the tower of the Hand. THATS why Tyrion is so much like Tywin (and would have made him proud if he wasnt born runtish) in so many ways while “his” other children dissapoint him too much.
"King Jaehaerys once told me that madness and greatness were two sides of the same coin. Every time a new Targaryen is born, he said, the gods toss the coin in the air and the world holds its breath to see how it will land."
Cercei = mad
Jamie = great but spolied through Cersei, his greatness becomes visible after his cleansing through Brienne.
Joffrey = are you seriously asking?
Myrcella = great (we see little of her but the way she found her way into the hearts of the Dornish looks like more than just an achievement of cuteness on a sweet little girl)
Tommen = Has a tendency towards madness if you ask me but hes too young we have too wait.