Carnot's theorem (conic sections)

6 points on the sides of a triangle with common conic section

The set of Carnot (after Lazare Nicolas Marguerite Carnot ) describes a relationship between conic sections and triangles .

In a triangle with points on the side , on the side and on the side , these six points lie on a common conic section if and only if the following equation applies: ${\displaystyle ABC}$ ${\displaystyle C_{A},C_{B}}$ ${\displaystyle AB}$ ${\displaystyle A_{B},A_{C}}$ ${\displaystyle BC}$ ${\displaystyle B_{C},B_{A}}$ ${\displaystyle AC}$

${\displaystyle {\frac {|AC_{A}|}{|BC_{A}|}}\cdot {\frac {|AC_{B}|}{|BC_{B}|}}\cdot {\frac {|BA_{B}|}{|CA_{B}|}}\cdot {\frac {|BA_{C}|}{|CA_{C}|}}\cdot {\frac {|CB_{C}|}{|AB_{C}|}}\cdot {\frac {|CB_{A}|}{|AB_{A}|}}=1}$.

With the help of partial ratios , this equation can also be noted as follows:

${\displaystyle (A,B;C_{A})\cdot (A,B;C_{B})\cdot (B,C;A_{B})\cdot (B,C;A_{C})\cdot (C,A;B_{C})\cdot (C,A;B_{A})=1}$

literature

• Huub PM van Kempen: On Some Theorems of Poncelet and Carnot . Forum Geometricorum, Volume 6 (2006), pp. 229-234.
• Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: With harmonious proportions to conic sections: pearls of classical geometry . Springer 2016, ISBN 9783662530344 , pp. 40, 168–173

Web links

• Carnot's theorem
• Carnot's Theorem for Conics on cut-the-knot.org