Any time I see this claim I want to bring up the fact that you can do it if the straightedge is marked in 2 places. See http://www.geom.uiuc.edu/docs/forum/angtri/ for a random link that explains the construction.
This is exactly the type of thing that has always made me think compass and straightedge constructions were even sillier than a lot of very obviously contrived mathematics.
That's an odd attitude for a math major to have. When geometers in ancient times thought mechanical curves were lesser second-rank objects, that was obviously a counterproductive prejudice that for a long time retarded the development of mathematics. But if you ask me, there is nothing "obviously contrived" about studying a configuration of lines and circles, their points of intersection, the lines and circles directly constructible from those intersections, and so on, ad infinitum. It strikes me as marvelously natural and beautiful and anything but contrived. That this corresponds to studying quadratic field extensions is a confirmation of that, if there were ever any serious doubt.
What I was saying is that I've always thought it was a silly restriction to say that you can't put a couple of arbitrary marks on your straightedge. It's just a matter of taste, I guess. The connection between these constructions and field theory is interesting, but I derive no pleasure from compass and straightedge proofs.
But you _can_ put _arbitrary_ marks on your straightedge. I haven't seen proofs needing or even using them, but you are free to place a new point at an arbitrary distance from a point A. You can then use the compass to transfer that distance to any line segment.
What you cannot do is put specific points on it, say three equidistant ones, or two at a distance of PI. If you allowed such constructs, the 'game' would become extraordinarily dull.
No, the rules specify "unmarked straightedge". And marking the straightedge is not allowed.
If you were allowed to mark the straightedge then you could put 2 random marks on it. With 2 random marks, you can trisect any angle. The location of the marks doesn't matter, just their existence.
It also, to me, isn't evident to me how one can proof that that Neusis construction is possible to place the ruler in that way without resorting to analytical means. I guess somebody must have thought about that. Are you aware of more rigid geometric proofs?
Well, the first problem is defining "line" and "circle" in your new topology, and this in itself tends to complicated not just the answer, but the question. For example, on the surface of a sphere, "lines" are a special case of circles, and all circles have two possible center points with two radii, which are distinct iff the circle is not a line. Once you've defined circles and lines, constructable sets looks very different. For example, you can always construct an absolute measure of distance, but the concept of parallel line segments doesn't exist, although you can have circles that are parallel to lines.
So can you trisect an angle in this slightly exotic topology? There's a good chance that you can, actually. There's a construction in the plane for trisecting a segment. If that construction is still valid on the sphere, you can take your angle and re-construct the legs with length equal to 1/4 of the sphere's circumference. Now, the line segment connecting the endpoints of these legs is an arc of a circle whose center is your angle. Trisecting this segment, and connecting the trisection back to the angle, will trisect the angle.
The notion of "angle" does not really make much sense in topology. However, one could ask this question in context of inner product spaces where one can define an angle in a natural way, but all real 2-dimensional inner product spaces are essentially all alike, so the answer is negative as well.
At least with a finite number of arcs and lines... true of anything I guess.
Actually, there are a lot of geometrical constructions that are possible with a finite number of operations with just a straightedge (not a ruler) and compass. Exactly bisecting an angle is easy. Trisecting is impossible.
See here I am a little confused. I never really understood why one couldn't do this:
construct an equilateral triangle atop the angle to be trisected
bisect any side, construct a new triangle with said 30deg angle giving a line segment 1/sqrt(3) of the original eq tri
bisect the new 60deg angle, constructing a new 30/60/90 tri
voila, 1/[sqrt(3)sqrt(3)] == 1/3. You have trisected a line with a compass and unmarked straightedge
This classical result holds in some generality, e.g. you also cannot trisect every angle with straightedge and compass in the hyperbolic plane (an exercise in Hartshorne's book Companion to Euclid).
