Carnot's theorem (conics)

Short description: A relation between conic sections and triangles

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 $ABC$ 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:

$\frac{|A{C}_A|}{|B{C}_A|}\cdot \frac{|A{C}_B|}{|B{C}_B|}\cdot \frac{|B{A}_B|}{|C{A}_B|}\cdot \frac{|B{A}_C|}{|C{A}_C|}\cdot \frac{|C{B}_C|}{|A{B}_C|}\cdot \frac{|C{B}_A|}{|A{B}_A|}=1$.

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