Is it permissible to claim to have met a girl at a public event (such as a wedding or seminar) in order to propose to her, so as to avoid her family's rejection due to their prior knowledge of your acquaintance via the internet, even though this might be considered a lie?

The questioner, male and female, should stay away from chatting in internet rooms, as it is impermissible, and they must repent to Allah for it. If the questioner's admiration for the woman's character is genuine and not based on whim, then proposing to her is the right thing to do, as the Prophet, peace and blessings be upon him, said: "Nothing is seen for two who love one another like marriage." There is no harm in being frank with one's family using allusions (ma'areedh), as they provide a way out of lying.