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This circle is called the incircle of the quadrilateral or its inscribed circle, its center is the incenter and its radius is called the inradius. Since these quadrilaterals can be drawn surrounding or circumscribing their incircles, they have also been called circumscribable quadrilaterals , circumscribing quadrilaterals , and circumscriptible ...
The inradius of the incircle in a triangle with sides of length , , is given by [7] = () (), where = (+ +) is the semiperimeter. The tangency points of the ...
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.
The radii and centers of these circles are called inradius and circumradius, and incenter and circumcenter respectively. From the definition it follows that bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclic quadrilaterals .
The inradius is = () (). The law of cotangents gives the cotangents of the half-angles at the vertices of a triangle in terms of the semiperimeter, the sides, and the inradius. The length of the internal bisector of the angle opposite the side of length a is [1]
The inradius (the radius of a circle inscribed in the rhombus), denoted by r, can be expressed in terms of the diagonals p and q as [10] = +, or in terms of the side length a and any vertex angle α or β as
where r is the inradius and R is the circumradius of the triangle. Here the sign of the distances is taken to be negative if and only if the open line segment DX (X = F, G, H) lies completely outside the triangle. In the diagram, DF is negative and both DG and DH are positive. The theorem is named after Lazare Carnot (1753–1823).
The radius of the sphere inscribed in a polyhedron P is called the inradius of P. Interpretations. All regular polyhedra have inscribed spheres, ...