What are some examples of Belle showing some form of intelligence in Beauty and the Beast? Why is she considered a nerd or a geek by so many people outside of fanpop?

I think people equate her with intelligence because she thinks for herself and isn't like the other young girls in the movie. Gaston even tells her "It's not right for a woman to read. Soon she starts getting *ideas*, and *thinking*..." Belle may not be a genius, but she is fairly intelligent because she wants to learn and she is curious. I do agree though that she isn't the smartest Disney princess. I think that would go to either Tiana or Rapunzel.
Ex. She reads Shakespeare which even in today's society is advanced literature.

Wanting to learn is typically what intelligent people do (not saying you're an idiot if you don't). But yes, Belle is not at all the smartest. She's PORTRAYED as being a genius next to the "dumb" townsfolk, and she's heavily marketed as the token "smart one" despite the fact that others are smarter than she is.

Context is everything. If compared to our society Belle is surely a well-opinionated, cultured bright girl, even if she doesn't show a genius level intelligence.

However, for the standards set in her own town, she's the most intelligent and "modern" woman. She's the only one to stop enough to try changing the way the Beast thinks and acts towards the world, she rebuffs brilliantly Gaston and while she looks as smart as the average girl, the "average girl" there is actually one of Gaston's grupies. And they're basically ditsies caring about looks and boys.

Thus, while Belle doesn't look that intelligent to us, she's a modern, well spoken woman, in a Victorian Age context