# Geometric constructions

Jump to: navigation, search

2010 Mathematics Subject Classification: Primary: 51M15 [MSN][ZBL]

The solution of certain geometric problems with the aid of various instruments (straight-edge, compasses, etc.), which are assumed to be perfectly accurate. The selection of the instruments determines the class of problems solvable by these means. The straight-edge and compasses are the two basic instruments for geometric constructions. A construction problem will be solvable with the aid of a straight-edge and compasses if the coordinates of the point sought can be written as expressions involving a finite number of the operations of addition, multiplication, division, and extraction of a square root, applied to the coordinates of given points (see, for example, Cyclotomic polynomials). If such expressions do not exist, the problem cannot be solved with the aid of a straight-edge and compasses. Such problems include, for example, duplication of the cube; trisection of an angle; and quadrature of the circle. Any construction problem that is solvable with the aid of compasses and a straight-edge can also be solved with the aid of a different set of instruments: the compasses alone (the Mohr–Mascheroni construction, G. Mohr, 1672; L. Mascheroni, 1797); a straight-edge with two parallel edges, which may be replaced by a triangle (A. Adler, 1890); a straightedge and a circle with a marked centre given in the plane of the figure (Poncelet–Steiner constructions, J.V. Poncelet, 1822; J. Steiner, 1833).