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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).
Some relations among the sides, incircle radius, ... where and are the circumradius and inradius respectively, and is the distance between the ...
By Euler's theorem in geometry, the distance between the circumcenter O and the incenter I is ¯ = (), 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.
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).
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:
Fuss' theorem gives a relation between the inradius r, the circumradius R and the distance x between the incenter I and the circumcenter O, for any bicentric quadrilateral. The relation is [1] [11] [22] + (+) =, or equivalently
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:
The distance between the circumcenter O and the orthocenter H is given by [2]: p. 19 =, where R is the circumradius. The squared distance from the incenter I to the orthocenter is [2]: p. 19 = +, where r is the inradius.