Inscribed figure

Familiar examples of inscribed figures include circles inscribed in triangles or regular polygons, and triangles or regular polygons inscribed in circles. A circle inscribed in any polygon is called its incircle, in which case the polygon is said to be a tangential polygon. A polygon inscribed in a circle is said to be a cyclic polygon, and the circle is said to be its circumscribed circle or circumcircle.

The inradius or filling radius of a given outer figure is the radius of the inscribed circle or sphere, if it exists.

The definition given above assumes that the objects concerned are embedded in two- or three-dimensional Euclidean space, but can easily be generalized to higher dimensions and other metric spaces.

For an alternative usage of the term "inscribed", see the inscribed square problem, in which a square is considered to be inscribed in another figure (even a non-convex one) if all four of its vertices are on that figure.

Properties

• Every circle has an inscribed triangle with any three given angle measures (summing of course to 180°), and every triangle can be inscribed in some circle (which is called its circumscribed circle or circumcircle).
• Every triangle has an inscribed circle, called the incircle.
• Every circle has an inscribed regular polygon of n sides, for any n ≥ 3, and every regular polygon can be inscribed in some circle (called its circumcircle).
• Every regular polygon has an inscribed circle (called its incircle), and every circle can be inscribed in some regular polygon of n sides, for any n ≥ 3.
• Not every polygon with more than three sides has an inscribed circle; those polygons that do are called tangential polygons. Not every polygon with more than three sides is an inscribed polygon of a circle; those polygons that are so inscribed are called cyclic polygons.
• Every triangle can be inscribed in an ellipse, called its Steiner circumellipse or simply its Steiner ellipse, whose center is the triangle's centroid.
• Every triangle has an infinitude of inscribed ellipses. One of them is a circle, and one of them is the Steiner inellipse which is tangent to the triangle at the midpoints of the sides.
• Every acute triangle has three inscribed squares. In a right triangle two of them are merged and coincide with each other, so there are only two distinct inscribed squares. An obtuse triangle has a single inscribed square, with one side coinciding with part of the triangle's longest side.
• A Reuleaux triangle, or more generally any curve of constant width, can be inscribed with any orientation inside a square of the appropriate size.