Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

, Carnot's theorem states that the sum of the

signed distance In mathematics and its applications, the signed distance function or signed distance field (SDF) is the orthogonal distance of a given point ''x'' to the boundary of a set Ω in a metric space (such as the surface of a geometric shape), with t ...

s from the

circumcenter In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcen ...

''D'' to the sides of an arbitrary triangle ''ABC'' is :

DF + DG + DH = R + r,\

where ''r'' is the

inradius In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

and ''R'' is the circumradius of the triangle. Here the sign of the

distances Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...

is taken to be negative if and only if the open

line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...

''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 Lazare Nicolas Marguerite, Comte Carnot (; 13 May 1753 – 2 August 1823) was a French mathematician, physicist, military officer, politician and a leading member of the Committee of Public Safety during the French Revolution. His military refor ...

(1753–1823). It is used in a proof of the

Japanese theorem for concyclic polygons Japanese may refer to: * Something from or related to Japan, an island country in East Asia * Japanese language, spoken mainly in Japan * Japanese people, the ethnic group that identifies with Japan through ancestry or culture ** Japanese diaspor ...

References

*Claudi Alsina, Roger B. Nelsen: ''When Less is More: Visualizing Basic Inequalities''. MAA, 2009, ,
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

* {{MathWorld, title=Carnot's theorem, urlname=CarnotsTheorem
Carnot's Theorem
at

cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet Union, Soviet-born Israeli Americans, Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow ...

Carnot's Theorem
by Chris Boucher. The

Wolfram Demonstrations Project The Wolfram Demonstrations Project is an Open source, open-source collection of Interactive computing, interactive programmes called Demonstrations. It is hosted by Wolfram Research. At its launch, it contained 1300 demonstrations but has grown t ...

. Theorems about triangles and circles