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Carnot's theorem (named after
''On Some Theorems of Poncelet and Carnot''
Forum Geometricorum, Volume 6 (2006), pp. 229–234. *Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: ''Mit harmonischen Verhältnissen zu Kegelschnitten: Perlen der klassischen Geometrie''. Springer 2016, {{ISBN, 9783662530344, pp. 40, 168–173 (German)
''Carnot's Theorem for Conics''
at cut-the-knot.org Theorems about triangles
Carnot's theorem (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 ...
) describes a relation between conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
s and triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
s.
In a triangle with points on the side , on the side and on the side those six points are located on a common conic section if and only if the following equation holds:
:.
References
*Huub P.M. van Kempen''On Some Theorems of Poncelet and Carnot''
Forum Geometricorum, Volume 6 (2006), pp. 229–234. *Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: ''Mit harmonischen Verhältnissen zu Kegelschnitten: Perlen der klassischen Geometrie''. Springer 2016, {{ISBN, 9783662530344, pp. 40, 168–173 (German)
External links
at cut-the-knot.org Theorems about triangles