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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).
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).
The inradius of the incircle in a triangle with sides of length , , is given by [7] = () (), where s = 1 2 ( a + b + c ) {\displaystyle s={\tfrac {1}{2}}(a+b+c)} is the semiperimeter. The tangency points of the incircle divide the sides into segments of lengths s − a {\displaystyle s-a} from A {\displaystyle A} , s − b {\displaystyle s-b ...
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
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 an acute triangle, the sum of the circumradius R and the inradius r is less than half the sum of the shortest sides a and b: [4]: p.105, #2690 + < +, while the reverse inequality holds for an obtuse triangle. For an acute triangle with medians m a, m b, and m c and circumradius R, we have [4]: p.26, #954
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]