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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).
If r and R are the inradius and the circumradius respectively, then the area K satisfies the inequalities [14] 4 r 2 ≤ K ≤ 2 R 2 . {\displaystyle \displaystyle 4r^{2}\leq K\leq 2R^{2}.} There is equality on either side only if the quadrilateral is a square .
1.2.3 Relation to area of the triangle. ... where and are the circumradius and inradius respectively, and is the distance between the ...
In geometry, Euler's theorem states that the distance d between the circumcenter and incenter of a triangle is given by [1] [2] = or equivalently + + =, where and denote the circumradius and inradius respectively (the radii of the circumscribed circle and inscribed circle respectively).
where r is the incircle radius and R is the circumcircle radius; hence the circumradius is at least twice the inradius (Euler's triangle inequality), with equality only in the equilateral case. [7] [8] The distance between O and the orthocenter H is [9] [10]
A complicated general formula is known for any number n of sides for the relation among the circumradius R, the inradius r, and the distance x between the circumcenter and the incenter. [5] Some of these for specific n are:
Carnot's theorem (inradius, circumradius), describing a property of the incircle and the circumcircle of a triangle; Carnot's theorem (conics), describing a relation between triangles and conic sections; Carnot's theorem (perpendiculars), describing a property of certain perpendiculars on triangle sides; In physics:
Examples of cyclic quadrilaterals. In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle.This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic.