compassandstraightedge

Compass and straightedge refers to the classical pair of geometric tools used for constructions in the Euclidean plane. A straightedge allows drawing a straight line through two known points, while a compass transfers measured distances and can draw a circle with a given center and radius. In the standard setup, a construction is a finite sequence of such draws beginning with a given set of points, with new points created at the intersections of lines and circles.

Typical constructions include drawing perpendicular bisectors and angle bisectors, constructing lines parallel to a given line, copying distances, and locating centers or intersections of circles. Using these operations one can construct regular polygons, divide segments into equal parts, and perform many standard geometric tasks. The theory of constructible figures emphasizes what can be achieved with these tools, and which points or lengths can be formed from a given starting data set.

Limitations arise from algebraic constraints on the numbers involved. Some historic problems, such as duplicating the cube, squaring the circle, and trisecting an arbitrary angle, have been proven impossible with just a straightedge and compass in general.

Key theoretical results describe how the two tools relate to the straightedge alone. The Mascheroni theorems (often stated as Mohr–Mascheroni) show that any straightedge-and-compass construction can be performed using a straightedge alone if a circle and its center are provided. The Poncelet–Steiner theorem strengthens this, showing that the same is true with only a circle and its center given. These theorems formalize the surprising power of a single circle in enabling complete constructions with a single tool.