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[Game of Thrones] Why did Tywin find Tyrion guilty?

Why did Tywin Lannister find Tyrion Lannister guilty of killing King Joffrey despite knowing Tyrion did not do it?
gpt
\- Tyrion inadvertently killed his mother by rupturing her birth canal with his malformed head. Tywin lost his only love. He hates Tyrion and blames him for it, he emotionally tortured him for years already.  


\- Tyrion is a misshapen dwarf, by the standards of Westeros, unfit to be a heir to the Lannister name. Tywin hated to lose a heir, but now that Jamie is out of Kingsguard due to injury, he can be Heir again, so Tyrion is politically useless. No need to keep him alive.  


\- the second reason for keeping Tyrion alive, his marriage to Sansa (and thus taking over the North) was nullified by her escape, and the fact that Boltons effectively took over the North anyway.  


\- Tyrion is as smart (if not smarter) than Tywin, but is a loudmouth arrogant wisecrack who disrespects his father and everything he stands for. Tywin *hates* being disrespected, because he remembers how his father, Tytos, allowed constant disrespect and it almost destroyed their House.  


\- Tywin actually does not know for sure that Tyrion is not involved somehow. At this point, he smartly suspects that it was either Olenna or the Martells, or maybe Sansa somehow, but he cannot rule out Tyrion's involvement.  


\- Tywin's hands are tied, sort of. Everybody in the court had heard Tyrion threaten and disrespect Jofferey, and Tyrion is almost universally hated by just about everyone except Littlefinger and Varys (both of whom have no way to clear his name or help him officially, and it would not even be their intention). No matter who is actually behind Joffs death, Cersei and the Court and even most of the common folk want the DEVIL MONKEY dead.