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Carnot's theorem (conics)

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Relation between conic sections and triangles For other uses, see Carnot's theorem (disambiguation).
6 points on the sides of triangle and their common conic section

Carnot's theorem (named after Lazare Carnot) describes a relation between conic sections and triangles.

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

| A C A | | B C A | ⋅ | A C B | | B C B | ⋅ | B A B | | C A B | ⋅ | B A C | | C A C | ⋅ | C B C | | A B C | ⋅ | C B A | | A B A | = 1 {\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} .

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)

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