Sqauring the circle...

[K^2]

There is a proof that this cannot be done. That's very different than lacking proof that it can be done. It's not a matter of us not having enough imagination or brain power to invent a method. We have figured out why it can't be done in general.

Simple example. Find an integer whose square is 7. I can easily show that no such integer exists. All I have to do is show that neither 1 * 1 nor 2 * 2 is equal to 7, because 3 * 3 = 9 > 7, and that means I can stop testing.

Your argument is equivalent to saying, "There are infinitely many integers. A human brain can't test them all, so we can't know that there isn't some other integer whose square is 7." But that's not how numbers work. If 1 * 1 = 1, and 2 * 2 = 4, then that's it. A square of absolutely any other integer will be too large to be equal to seven, and I don't have to test them all.

Pi is not algebraic, therefore it is not constructible, and therefore squaring the circle is impossible. I know that you don't understand what any of these things actually mean, but people who studied a little bit of mathematics absolutely do. And it's equivalent to excluding all of the infinite possible constructions. You aren't going to find a construction that works, because to find one would be to prove that pi is algebraic. That's like proving that it's exactly square root of 12, or something like that. Which it isn't. Again, we have a proof that it's not. It's just like the situation with all other numbers being too big. All other construction methods just don't fit.