noncircumscribed

Noncircumscribed is not a standard term in geometry. When it is used, it is typically as an informal way to describe a figure that does not exhibit the common circle-related properties that many polygons can have. The two conventional properties are captured by the terms cyclic and tangential.

A polygon is cyclic if all its vertices lie on a single circle; in other words, the polygon has a circumcircle. A polygon is tangential if there exists a circle that is tangent to all of its sides, that is, the polygon has an incircle. These two properties describe different ways a figure can relate to a circle, and they are independent: a polygon can be cyclic, tangential, both, or neither.

Therefore, a noncircumscribed polygon is typically understood as a polygon that is neither cyclic nor tangential. For example, many irregular quadrilaterals are not cyclic (no single circle through all four vertices) and not tangential (no circle tangent to all four sides). Some polygons may be cyclic but not tangential, some tangential but not cyclic, and some neither.

In practice, the term noncircumscribed is rarely used in formal discussions. Geometric discussions prefer precise classifications, such as describing a polygon as cyclic, tangential, or neither. When clarity is required, it is better to specify whether a figure has a circumcircle, an incircle, both, or neither, rather than using the umbrella term noncircumscribed.

See also: cyclic polygon, tangential polygon, incircle, circumcircle, circumscribed and inscribed terminology.