Euclides

Compass-and-straightedge or ruler-and-compass construction is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass.
The idealized ruler, knownas a straightedge, is assumed to be infinite in length, and has no markings on it and only one edge. The compass is assumed to collapse when lifted from the page, so may not be directly used totransfer distances. (This is an unimportant restriction, as this may be achieved via the compass equivalence theorem.) More formally, the only permissible constructions are those granted by Euclid's firstthree postulates.
Every point constructible using straightedge and compass may be constructed using compass alone. A number of ancient problems in plane geometry impose this restriction.
The mostfamous straightedge-and-compass problems have been proven impossible in several cases by Pierre Wantzel, using the mathematical theory of fields. In spite of existing proofs of impossibility, some persistin trying to solve these problems.[1] Many of these problems are easily solvable provided that other geometric transformations are allowed: for example, doubling the cube is possible using geometricconstructions, but not possible using straightedge and compass alone.
Mathematician Underwood Dudley has made a sideline of collecting false ruler-and-compass proofs, as well as other work bymathematical cranks, and has collected them into several books.
Compass and straightedge tools
The "compass" and "straightedge" of compass and straightedge constructions are idealizations of rulers andcompasses in the real world:
* The compass can be opened arbitrarily wide, but (unlike some real compasses) it has no markings on it. Circles can only be drawn using two existing points which givethe centre and a point on the circle. The compass collapses when not used for drawing, it cannot be used to copy a length to another place.
* The straightedge is infinitely long, but it has no...