Carnot's theorem (conics) explained

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

In a triangle

ABC with points C _{A,} C _{B} on the side AB , A _{B,} A _{C} on the side BC and B _{C,} B _{A} on the side AC those six points are located on a common conic section if and only if the following equation holds: (|AC _{A|})/(|BC _{A|}) ⋅ (|AC _{B|})/(|BC _{B|}) ⋅ (|BA _{B|})/(|CA _{B|}) ⋅ (|BA _{C|})/(|CA _{C|}) ⋅ (|CB _{C|})/(|AB _{C|}) ⋅ (|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,, pp. 40, 168–173 (German)

External links

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