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

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

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)

External links

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