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

, Carnot's theorem states that the sum of the

signed distance In mathematics and its applications, the signed distance function (or oriented distance function) is the orthogonal distance of a given point ''x'' to the boundary of a set Ω in a metric space, with the sign determined by whether or not ''x'' ...

s from the

circumcenter In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

''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 In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

of the triangle. Here the sign of the

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

is taken to be negative if and only if the open

line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...

''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, Count Carnot (; 13 May 1753 – 2 August 1823) was a French mathematician, physicist and politician. He was known as the "Organizer of Victory" in the French Revolutionary Wars and Napoleonic Wars. Education and early ...

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

Japanese theorem for concyclic polygons __notoc__ In geometry, the Japanese theorem states that no matter how one triangulates a cyclic polygon, the sum of inradii of triangles is constant.Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007 (orig. 1929). Conversel ...

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-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...

Carnot's Theorem
by Chris Boucher. The

Wolfram Demonstrations Project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...

. Theorems about triangles and circles