Inscribed and Circumscribed Figures

in elementary geometry. A polygon is inscribed in a convex curve and a curve is circumscribed about a polygon if all the vertices of the polygon lie on the curve. A polygon is circumscribed about a curve and the curve is inscribed in a polygon if every side of the polygon or its extension is tangent to the curve. The curve is most often a circle. Every triangle has one circumscribed and one inscribed circle. A convex quadrilateral has a circumscribed circle if and only if the sum of its opposite angles equals 180°. In order for a quadrilateral to have an inscribed circle, it is necessary and sufficient that the sum of the lengths of one pair of opposite sides equal the sum of the lengths of the other pair. A polygon can be inscribed in a circle if this latter property belongs to the quadrilaterals formed by a diagonal of the polygon and three sides and also if the perpendicular lines drawn through the centers of the sides intersect at one point. An inscribed circle exists if and only if the bisectors of the interior angles of a polygon intersect at one point. In projective geometry an important role is played by theorems about a hexagon inscribed in and circumscribed about a conic section.

Inscribed and circumscribed figures are also considered in space. In this case polyhedrons are used instead of polygons and a convex surface (most often a sphere) instead of a convex line. It is also possible to speak of a cone or cylinder inscribed in a sphere, of a sphere inscribed in a cone, and so on.