“Carnot's theorem”的意思、由来-开放百科全书

词条

Carnot's theorem

释义

References
External links

{{For|the similarly named theorem in thermodynamics|Carnot's theorem (thermodynamics)}}

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

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 9780883853429S, p.[https://books.google.de/books?id=U1ovBsSRNscC&pg=PA99 99]
Frédéric Perrier: Carnot's Theorem in Trigonometric Disguise. The Mathematical Gazette, Volume 91, No. 520 (March, 2007), pp. 115-117 ([https://www.jstor.org/stable/40378302 JSTOR])
David Richeson: [https://www.maa.org/press/periodicals/the-japanese-theorem-for-nonconvex-polygons-a-japanese-temple-problem The Japanese Theorem for Nonconvex Polygons - Carnot's Theorem]. Convergence, December 2013

External links

{{MathWorld|title=Carnot's theorem|urlname=CarnotsTheorem}}
Carnot's Theorem at cut-the-knot
Carnot's Theorem by Chris Boucher. The Wolfram Demonstrations Project.

2 : Triangle geometry|Theorems in plane geometry

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。