A doubt about what is straightedge and compass construction The doubt comes from a try. To bisect a segment, I wonder whether I can first choose an arbitrary degree of opening for the compass, then use the compass to measure the segment until I can measure it with two times of use of the compass. It seems that we get the bisection now. Is this foolish method a rigorous straightedge and compass construction? And why?
I tend to believe it is wrong. Because once you allow such 'try method', I think then those impossibility in straightedge and compass construction will become possible, that is horrible. But I actually don't know why it's wrong.
 A: The issue here is twofold: a compass and straightedge construction must occur in finitely many steps and it must constitute an actual geometric proof.

*

*You haven't shown that your procedure of "try a radius then try again" will terminate in finitely many steps. Perhaps you can do some sort of iterative method so that your sequence of guesses for the radius converges to half the length of your line segment, but this is no longer a compass and straightedge proof since those must be done in finitely many steps.


*Suppose by some happenstance your method does, in finitely many steps, get you precisely the radius you want. How do you verify that this is actually correct? You arrived at this radius by guessing but even if you randomly get the right answer, you need to prove that it is so. You could say that you take your compass with the correct radius $r$ and draw one circle starting at $a$ and another at the intersection of the line segment $\overline{ab}$ with this circle of radius $r$. Since we're assuming you did get the right radius, it will be the case that this second circle actually intersects $b$, but you cannot actually verify this. What if you are actually some tiny length like $10^{-TREE(3)}$ away from $b$? the exit condition of your procedure is "when I get the right radius" but since your procedure does not constitute a proof, you can only ever guess at when the radius looks right. The beauty of these constructions is that they are utterly ambiguous. A valid compass and straightedge construction is 100% precise without worrying about imprecise guesses. That's why the straightedge is not a ruler with markings! We don't allow measurements that are not provable from our idealized tools.
To explain this second point some more, let me give you a correct construction of the bisection of a line segment $\overline{ab}$. Put one leg of your compass on $a$ and another on $b$, the rotate your compass around $a$ to form a circle $C_1$. This is one of the allowed moves of this game, since the points $a, b$ are assumed to be known to us. Then, put your compass in the same position but rotate around $b$ to get another circle $C_2$. These circles intersect in precisely two points, say $c, d$. This is not from us eyeballing - it is provably so! Now, we know of the points $c, d$ so draw the line segment $\overline{cd}$. Again, drawing a line between known points is one of our legal moves. The intersection of $\overline{cd}$ and $\overline{ab}$ will precisely bisect $\overline{ab}$. If my dictation wasn't clear, look at this link. You can press z and x to go back and forth.
Now I have not actually given a formal geometric proof of this construction, but my point is that the I could translate it to one should I choose to do so. I could prove rigorously that my circles $C_1$ and $C_2$ intersect in precisely two points and I could prove that $\overline{cd} \cap \overline{ab}$ forms the bisection point. I did all of this in finitely many steps and I never had to guess to say "well this looooks close enough to the bisection." I can precisely prove using the formal definitions of compasses and straightedges (which come from Euclid's postulates) that the point I arrive at is the bisection point.
EDIT: I'll answer your comment in here, since it's too long to fit in a response.
That's a fair objection, and I think the only ways to counter it are through formalism and philosophy. For the formal viewpoint, compass and straightedge constructions are visual representations of proofs from Euclid's postulates, which I linked in my answer. The steps I give like "put a leg on $a$" are representatives of postulates like the existence of a circle with a certain center and radius. Then by a "precise" construction I mean one following from these postulates, which yours does not. Also, a proof is necessary a finite sequence of deductions so any classical construction must therefore also be finite. This is the formal difference between our approaches, but it's not very motivating beyond "well Euclid says so." You can ask more deeply why we choose to take these postulates as "precise." I think this inevitably begs questions of history and philosophy. I would also like to point out that I am much more familiar with the formalism than the philosophy, but I'll try my best to get across some ideas and, more importantly, bring up some relevant terms.
I think it is best to try to start by asking why the Greek's were so interested in compasses and straightedges. After all, the story goes that at the door of Plato's academy there was an inscription reading "Let no one ignorant of geometry enter." Whether or not this is apocryphal[1] is not really the point; it is merely illustrative of the philosophical importance the Greeks placed upon geometry. A large part of this was due to the theory of forms[2]. Roughly (which is how well I understand it...), forms are immaterial, perfect, ideal objects. For instance, a circle I draw with an actual physical compass is just a shadow of the ideal form of a circle[0]. In the Republic[3], Plato said that

[geometers] make use of the visible forms and talk about them, though they
are not thinking of them but of those things of which they are
a likeness, pursuing their inquiry for the sake of the square as
such and the diagonal as such, and not for the sake of the image
of it which they draw

So to the geometers at the time of Plato, a compass and straightedge construction was not really just the physical sequences of moves (I put this leg here and rotate about this, I draw a line from here to here, etc). These were a way to grapple with the ideal geometry forms by using "the image of it which they draw." If you're familiar with Plato's allegory of the cave this may ring a bell.
What I mean to say by all of this is that as I understand it, these compass and straightedge constructions arise from an attempt to understand these precise objects like circles and lines and points via their imprecise drawings and likenesses in the real world. When I say that your "guess until your touch $b$" method is imprecise, I suppose I am saying that that is an instance of reasoning with the "shadows on the wall" of the ideal forms we're trying to understand. When I invoke one of Euclid's postulates, I am making a statement about the ideal objects, and when I draw using an actual compass or straightedge, I am merely "[making] use of the visible forms" while "not thinking of them but of those things of which they are a likeness." Your approach of "guess until you touch $b$" is a different type of thought to what the Greek geometers were trying to do. It is working directly with your imperfect drawings rather than using the drawings as a proxy for this higher, immaterial truth the Greeks were after.
This is all very highfalutin, and I am far from knowledgeable in this realm, but I hope my summary and references will help you understand a bit more why your construction is not classically valid. I suppose to sum it up, when you make an imprecise statement like "put your compass leg on $a$" it is because you are using images to deal with these immaterial forms. However, when you have to guess you are only able to do this for the image itself. The idealized forms of these constructions are perfectly precise. The construction I describe is unquestionably the bisection. What I draw by hand or by computer is an imprecise mental model of that perfect precision, whereas the imprecision of guessing is reflective only of the shadows, not the forms themselves.
I hope my rambling helped. Please feel free to discuss this with me further, and here are some of the articles I referred to:
https://topologicalmusings.wordpress.com/2008/01/18/platonic-idealism-and-geometry/ [0]
https://www.dialogues-de-platon.org/faq/faq009.htm [1]
https://en.wikipedia.org/wiki/Theory_of_forms [2]
https://link.springer.com/book/10.1007/978-1-4612-0803-7 [3]
