Carnot's theorem (conics)
{{short description|Relation between conic sections and triangles}}
{{other|Carnot's theorem (disambiguation)}}
Carnot's theorem (named after Lazare Carnot) describes a relation between conic sections and triangles.
In a triangle with points on the side , on the side and on the side those six points are located on a common conic section if and only if the following equation holds:
:
\frac
| AC_A |
| BC_A |
| AC_B |
| BC_B |
| BA_B |
| CA_B |
| BA_C |
| CA_C |
| CB_C |
| AB_C |
| CB_A |
| AB_A |
.
References
- Huub P.M. van Kempen: [http://forumgeom.fau.edu/FG2006volume6/FG200626.pdf 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
- [http://users.math.uoc.gr/~pamfilos/eGallery/problems/CarnotConics.html Carnot's theorem]
- [http://www.cut-the-knot.org/triangle/CarnotForConics.shtml Carnot's Theorem for Conics] at cut-the-knot.org