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Geometric Constructions

The notion of  geometric constructions go back at least to Greek antiquity. The classical constructions are called Euclidean constructions, but they certainly were known prior to Euclid. The only instruments allowed to be used in these constructions are compass and straightedge. The compass is used to establishes the equidistance, and the straightedge establishes the collinearity. Thus Euclidean geometric constructions are based on these two procedures:

• The compass is anchored at a center point, and keeps the drawing instrument at a fixed distance from that point. Thus the points on the curve (circle) drawn by a compass are equidistant from the center point.
• The straightedge is used only for drawing lines. It is often called straightedge, but no measurements are allowed in Euclidean constructions. This instrument is used to draw the straight line passing through two given points.

Constructions are understood as a theoretical exercise, whether for instance a line has zero width, and various  physical imperfections of the drawing instruments are neglected.

Marked straightedges and protractors are not allowed in the classical Euclidean constructions. Even if the marked straightedge is not used in Euclid’s Elements, other Greek geometers used it, for instance Hippocrates of Chios (ca. 430 BC). Pappus reports that Applonius of Perga  (ca. 262-190 BC) wrote two books on constructions using marked straightedge.

The constructions using marked straightedge were called neusis constructions by ancient Greeks. [1]

When the straightedge and compass constructions did not offer a solution then also other types of construction instruments were accepted, for instance

• constructions that in addition to this use conical sections (ellipses, parabolas, hyperbolas)
• constructions that in addition to this use other curves,  as the so-called conchoid curve (Nicomes used it to solve the problem of trisecting the angle)

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