Everything you can do with a ruler and a compass you can do with the compass alone.
Well, not everything. For example, you can't draw straight lines using a compass. There is no
talking about it. However, you can do everything reasonable. I hope you would find this claim no less
remarkable.
In what is known as the Geometry of Compass, a straight line is defined by any pair
of two points. Starting with two points, other points can be constructed with compass alone. Thus in the
following constructing a straight line means finding two points that belong to that line.
Remark
There are geometries in which the ruler is never used to start with. E. g., in finite geometries that only contain a finite number of points and lines, a line is just a (finite) collection of points. On the sphere, the role of straight lines is played by the great circles. The question of geometric construction with the compass alone is not concerned with such kinds of geometries. Geometry of Compass only deals with constructions in the Euclidean plane, and its basic question could be formulated as, What ruler-and-compass constructions could be accomplished with the compass alone?
The assertion that every ruler-and-compass construction could be accomplished with a compass is due to Lorenzo Mascheroni (1750-1800) and appeared in his 1797 tractate The Geometry of Compasses.
Interestingly, in 1928 the Danish mathematician Hjelmslev discovered in a bookshop in Copenhagen a book
by G. Mohr titled Euclides Danicus (The Danish Euclid) and published in 1672 in Amsterdam. To his great surprise Hjelmslev found a complete treatment of the Mascheroni result in the first part of the book. For this reason, constructions with compass only are commonly referred to as the Mohr-Mascheroni constructions.
With regard to the Mascheroni's result, instead of checking every single construction in the plane we
agree that such constructions can be accomplished with a sequence of the four basic ones:
To draw a circle with the given center and radius
To find the point of intersection of two circles
To find the points of intersection of a straight line and a circle
To find a point of intersection of two straight lines
The difficulty obviously lies with the last two problems. In the Geometry of Compass
constructions may be awfully obscure even for simple problems. To avoid complicating the matters it's
always useful to split a problem into a number of simpler steps. A proof to the Mascheroni result will
emerge as a combination of the problems below. (However, not all of the problems are related to the
proof.)
Problems (Use a compass only)
In all problems below a segment AB is given by its end points A and B.
