Carnot's theorem (inradius, circumradius)

Carnot's theorem (inradius, circumradius) For other uses, see Carnot's theorem (disambiguation). {displaystyle {begin{aligned}&DG+DH+DF\={}&|DG|+|DH|-|DF|\={}&R+rend{aligned}}} 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. The Mathematical Gazette, Volume 91, No. 520 (March, 2007), pp. 115–117 (JSTOR) David Richeson: The Japanese Theorem for Nonconvex Polygons – Carnot's Theorem. Convergence, December 2013 External links Weisstein, Eric W. "Carnot's theorem". MathWorld. Carnot's Theorem at cut-the-knot Carnot's Theorem by Chris Boucher. The Wolfram Demonstrations Project. Categories: Theorems about triangles and circles

Si quieres conocer otros artículos parecidos a Carnot's theorem (inradius, circumradius) puedes visitar la categoría Theorems about triangles and circles.

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