Final answer:
Sophie knew Agatha was her nemesis by looking into her eyes.
Step-by-step explanation:
In the given text, Sophie knew that Agatha was her nemesis by looking into her eyes. The mention of Sophie finding it strangely embarrassing to meet Fay's violet eyes suggests that Sophie saw something in Agatha's eyes that confirmed she was her nemesis.
This is a common literary device called 'the eyes as a window to the soul,' where a character's true intentions or personality are revealed through their eyes.
For example, in Shakespeare's play 'Macbeth,' Lady Macbeth's guilt and remorse are shown through her haunted eyes.