circumscris

Circumscris is a term used in Romanian mathematical vocabulary to describe a figure that is circumscribed around a circle, meaning the figure has an incircle that is tangent to all its sides. In English, the corresponding concept is typically described as “circumscribed about a circle” or as a “tangential” polygon.

Etymology and form: The term derives from Latin circum- meaning around and scribere meaning to write, through the Romanian verb circumscrie (to circumscribe). Circumscris is the masculine past participle form used to modify a noun, for example circumscris polygon or circumscris about a circle.

Definition and scope: A convex polygon circumscris about a circle is one for which there exists a circle tangent to every side; the circle is called the incircle and its center is the incenter. Such polygons are commonly referred to in English as tangential polygons. Not all polygons are circumscris; a necessary and sufficient condition depends on the existence of tangent lengths from each vertex to the incircle. For triangles, an incircle always exists, so every triangle is circumscris. For quadrilaterals and higher polygons, the existence of an incircle is a special property.

Examples and properties: Regular polygons, such as equilateral triangles and squares, are circumscris because they possess an incircle. In general, a polygon that is circumscris has equal sums of lengths of opposite sides in the quadrilateral case, and a more complex set of conditions applies to polygons with more than four sides. The concept is closely related to, but distinct from, polygons that are inscribed in a circle (cyclic polygons).

See also: Circumscribed circle, incircle, tangential polygon, circumscription.