Rulerandcompass

Ruler and compass refers to the traditional combination of tools used in Euclidean geometry: the straightedge (unmarked ruler) for drawing lines and the compass for drawing circles and transferring distances. Together, they enable geometric constructions by creating new points at the intersections of lines and circles.

Common constructions include drawing a line through two given points, constructing perpendicular bisectors and angle bisectors, locating intersection points, constructing regular polygons, and drawing tangents to circles. These techniques underlie Euclidean geometry and have long informed geometric drafting in engineering and architecture.

From a theoretical standpoint, any construction achievable with a ruler and compass can be carried out with these tools in the same framework. Classical results address the limitations of the tools: several problems are provably unsolvable in general with ruler and compass alone, notably duplicating the cube, squaring the circle, and trisecting a general angle.

Two important theorems about constructibility relate to the tools themselves. Mascheroni's theorem states that any ruler-and-compass construction can be performed using only a compass. The Poncelet–Steiner theorem shows that, given a circle and its center, all ruler-and-compass constructions can be done with just a straightedge.

Historically, the ruler and compass approach dates to ancient Greek geometry and is central to Euclid’s Elements. In modern contexts, the concepts remain foundational in education and in the understanding of geometric reasoning, even as digital tools increasingly handle precise construction in practice.