All these three problems are my own and I hope you liked them. For reasons I shall not say I cannot fully respond to all three answers right now, but soon will.
1)For the "perimecenter", I solved it by cheating, I used Ceva's Theorem for coincidence. Use it as a hint, before I respond.
2)This one is simple, when you draw a line as in #1 to create equal area, the heights of the two triangle are the same. That means the bases must be equal for there to be equal area. Thus, that line is acutally a median. And it is a known fact that all medians pass through a common point. Again you can use Ceva's Theorem.
3)The physicist is assuming that the new number that he formed from using the diagnol argument is contained within the set of rationals. He never confirmed this. In the original diagnol argument the number formed has a decimal expansion and is therefore real and is thus contained in the set. The same cannot be said here for it is not contained within the set.