The New International Encyclopædia/Construction

CONSTRUCTION (Lat. constructio, from construere, to construct, from com-, together + struere, to heap). In geometry, the process of drawing a figure so as to satisfy the conditions of the given problem. Thus, to construct an equilateral triangle of side a: with each end of a as a centre and with a as a radius, describe a circle; connect either intersection with the ends of a. Here the construction is not unique, since two triangles satisfy the condition. In solving problems a valuable method is to assume the construction and investigate the properties of the figure. Thus, to draw a line through a given point parallel to a given line: assuming the construction and a transversal of the parallels through the given point, it appears that the alternate angles are equal; hence, to construct the figure, draw a line through the point cutting the given line and construct the alternate angle.

Another fruitful method is that of the intersection of loci; e.g. if it is known that a point is on each of two intersecting straight lines, it is uniquely determined at their point of intersection; but if it is on a straight line and a circumference which the line intersects, it may be either of the two points of intersection.

The best works upon the constructions of elementary geometry are Petersen, Methods and Theories (Copenhagen and London, 1879); Rouché and de Comberousse, Traité de géométrie (Paris, 1900); and Alexandroff, Problèmes de géométrie élémentaire, translated into French by Aitoff (Paris, 1899). Consult also Beman and Smith, New Plane and Solid Geometry (Boston, 1899).