As a student, I was taught that the Jordan curve theorem is a great example of an intuitively clear statement which has no simple proof.


What is the simplest known proof today?


Is there an intuitive reason why a very simple proof is not possible?



Nice proof of the Jordan curve theorem?
There's a short proof (less than three pages) that uses Brouwer's fixed point theorem, available here:


[The Jordan Curve Theorem via the Brouwer Fixed Point Theorem](http://www.maths.ed.ac.uk/~aar/jordan/maehara.pdf)


The goal of the proof is to take Moise's "intuitive" proof and make it simpler/shorter. Not sure whether you'd consider it "nice," though.


