Carnot's theorem
Encyclopedia
In Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, Carnot's theorem, named after Lazare Carnot
Lazare Carnot
Lazare Nicolas Marguerite, Comte Carnot , the Organizer of Victory in the French Revolutionary Wars, was a French politician, engineer, and mathematician.-Education and early life:...

 (1753–1823), is as follows. Let ABC be an arbitrary triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

. Then the sum of the signed distances from the circumcenter D to the sides of triangle ABC is


where r is the inradius and R is the circumradius. Here the sign of the distances is taken negative if and only if the line segment
Line segment
In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...

 DX (X = F, G, H) lies completely outside the triangle. In the picture DF is negative and both DG and DH are positive.

Carnot's theorem is used in a proof of the Japanese theorem for concyclic polygons.

External links

  • Carnot's Theorem at cut-the-knot
    Cut-the-knot
    Cut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...

  • Yet another Carnot's Theorem with multiple applications at cut-the-knot
    Cut-the-knot
    Cut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...

  • Carnot's Theorem by Chris Boucher. The Wolfram Demonstrations Project
    Wolfram Demonstrations Project
    The Wolfram Demonstrations Project is hosted by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience. It consists of an organized, open-source collection of small interactive programs called Demonstrations, which are meant to visually and...

    .
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