how dose he reach the conclusion about who killed welligton

After Christopher's father gives him a bath, he is very open with Christopher and says that he can trust him and that he is sorry. He says that he was only lying to protect him, but that from now on he would always be honest. He tells Christopher that he was the one who killed Wellington. He and Eileen had been getting on really well after his mother left and he thought that one day they might all move in together, but he and Eileen had fights. After one really horrible one where she said horrible things, she threw him out and the dog came after him, and that's when all the bad feelings over the last two years came bubbling up.