Final answer:
The line that illustrates Maître Hauchecorne's conflict in "A Piece of String" is about the shame he feels over being falsely accused, highlighting his internal struggle and the value he places on his reputation.
Step-by-step explanation:
The line from "A Piece of String" that illustrates Maître Hauchecorne's conflict is: "There is nothing so shameful as to be placed under a cloud on account of a lie." This sentence encapsulates the essence of Hauchecorne's struggle with being wrongly accused and the shame and frustration he experiences due to this accusation. This line reveals the character's inner turmoil and the importance he places on his reputation and honesty.