circumscribe

Circumscribe is a verb meaning to surround or enclose on all sides. In geometry, it describes the act of drawing a figure around another so that the outer figure either passes through all the inner figure’s vertices or is tangent to the inner figure. The word comes from Latin circum- “around” and scribere “to write,” with the sense of drawing around.

A common geometric use is the circumcircle. The circumcircle of a polygon is the circle that passes through all its vertices. The polygon is called cyclic when it has a circumcircle. Every triangle has a circumcircle, and the center of that circle is the circumcenter, found at the intersection of the perpendicular bisectors of the sides. Not all polygons are cyclic; the existence of a circumcircle depends on the alignment of its vertices.

Another standard usage describes a polygon circumscribed about a circle, meaning all its sides are tangent to the circle. The circle in this case is the incircle, and the polygon is said to be tangential. The center of the incircle is the incenter, the intersection point of the polygon’s angle bisectors. A triangle can be circumscribed about a circle, with the circle inscribed in the triangle.

In broader language, circumscribe can simply mean to enclose or limit something within bounds, but in geometry it precisely indicates the relationship of a figure surrounding or being surrounded by another as described above.