Refuting the Flat-Earth theory

[FlatAssembler]

It would be useless to address some bad idea rationally when the person who holds it is fully willing to declare that antarctica is part of the conspiracy.

You are right. Perhaps I should add something like "If you are willing to claim that Antarctica is a part of a conspiracy, do you realize there are plane routes flying from Australia to South America over the Antarctica?" to the "polar day and polar night on Antarctica" argument in the OP.