Given a segment length and , we can't construct , , or , but we can do by Pythagoras.
We can't construct , , but we can do by drawing semicircle on and draw a line perpendicular to the point these 2 segments intersect. The line from circumference and the intersection has the length
What I see is that the degree of what we want to construct tells you if it can be constructed or not. Degree is the exponent of any length. It can be constructed iff the degree of given length is equal to degree of length we want to construct. (Note: the description is not fully written yet, I have too much homework to do XD.)
For example, given length , have degree , but has degree , and also (like polynomials, adding polynomials only consider the highest degree of result). And then, has degree 1. So we can construct it.
Another example, given length , have degree , , have degree because . But has degree , so we can construct it.
Are there any proofs or, if you wish, counterexamples of this thing I don't even have an idea what to call. It just popped out of my mind yesterday. XD
Maybe the degree is the invariance of construction? I don't know. :3