Carnot's theorem (inradius, circumradius)

Carnot's theorem (inradius, circumradius) Pour d'autres usages, see Carnot's theorem (désambiguïsation). {style d'affichage {commencer{aligné}&DG+DH+DF\={}&|DG|+|DH|-|DF|\={}&R+rend{aligné}}} In Euclidean geometry, Carnot's theorem states that the sum of the signed distances from the circumcenter D to the sides of an arbitrary triangle ABC is {displaystyle DF+Dg+DH=R+r, } 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). It is used in a proof of the Japanese theorem for concyclic polygons.

References Claudi Alsina, Roger B. Nelsen: When Less is More: Visualizing Basic Inequalities. MAA, 2009, ISBN 978-0-88385-342-9, p.99 Frédéric Perrier: Carnot's Theorem in Trigonometric Disguise. La gazette mathématique, Le volume 91, Non. 520 (Mars, 2007), pp. 115–117 (JSTOR) David Richeson: The Japanese Theorem for Nonconvex Polygons – Carnot's Theorem. Convergence, Décembre 2013 Weissstein externe gauche, Eric W. ">

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