straightedgeonly

Straightedgeonly (or straightedge-only) is a term used to describe performing geometric constructions using a straightedge alone under certain fixed conditions. In classical Euclidean geometry, constructions are typically carried out with both a straightedge and a compass. The concept of straightedge-only constructions shows that, with appropriate auxiliary information, many of the same results can be obtained using only a straightedge.

The core result is captured by the Poncelet–Steiner theorem. It states that if a single circle together with its center is provided in the plane, then every construction that is possible with a straightedge and compass can be performed with a straightedge alone. The theorem highlights how extra data can compensate for the lack of a compass, enabling the reconstruction of lengths, angles, and other geometric relations through projective and metric techniques.

Historically, the theorem is attributed to work by Jean-Victor Poncelet and Jakob Steiner in the 19th century, and it remains a foundational result in the theory of geometric constructions. It situates straightedge-only methods within a broader framework that also includes compass-only constructions (the Mohr–Mascheroni theorem) and classical straightedge-and-compass constructions.

In practice, straightedge-only constructions are discussed in the context of geometry education, mathematical logic, and computer-aided geometry, where the feasibility and limits of using restricted tools are analyzed. They illustrate how auxiliary objects and projective properties can substitute for direct metric tools in certain problems.

See also: straightedge and compass, Mohr–Mascheroni theorem, constructibility, projective geometry.