Difference between revisions of "Chasles theorem"
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| − | If $A,B,C$ are three arbitrary points on a straight line, then $\overline{AB}+\overline{BC}=\overline{AC}, where $\overline{AB},\overline{BC},\overline{AC}$ are the lengths of the directed line segments. Chasles' theorem can be generalized to the case of the surface of oriented triangles and the volumes of oriented tetrahedra (see ). | + | If $A,B,C$ are three arbitrary points on a straight line, then $\overline{AB}+\overline{BC}=\overline{AC}$, where $\overline{AB},\overline{BC},\overline{AC}$ are the lengths of the directed line segments. Chasles' theorem can be generalized to the case of the surface of oriented triangles and the volumes of oriented tetrahedra (see ). |
A motion of the first kind (orientation-preserving), different from a rotation and a translation, is the product of a translation and a rotation the axis of which is parallel to the direction of the translation (a so-called screwing motion). The theorem was proved by M. Chasles in 1830. | A motion of the first kind (orientation-preserving), different from a rotation and a translation, is the product of a translation and a rotation the axis of which is parallel to the direction of the translation (a so-called screwing motion). The theorem was proved by M. Chasles in 1830. | ||
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====Comments==== | ====Comments==== | ||
| − | Any book on linear algebra and analytic geometry can serve as a reference, since both theorems are easy exercises. Another result that also is called Chasles' theorem can be found in [[#References|[a1]]]: If the polars of the vertices of a triangle (cf. [[ | + | Any book on linear algebra and analytic geometry can serve as a reference, since both theorems are easy exercises. Another result that also is called Chasles' theorem can be found in [[#References|[a1]]]: If the polars of the vertices of a triangle (cf. [[Polar]]) do not coincide with the respectively opposite sides, then they meet these sides in three collinear points. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H.S.M. Coxeter, "Projective geometry" , Blaisdell (1964)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> P.S. Modenov, "Analytic geometry" , Moscow (1969) (In Russian)</TD></TR> | ||
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> H.S.M. Coxeter, "Projective geometry" , Blaisdell (1964)</TD></TR> | ||
| + | </table> | ||
Latest revision as of 14:01, 30 April 2023
If $A,B,C$ are three arbitrary points on a straight line, then $\overline{AB}+\overline{BC}=\overline{AC}$, where $\overline{AB},\overline{BC},\overline{AC}$ are the lengths of the directed line segments. Chasles' theorem can be generalized to the case of the surface of oriented triangles and the volumes of oriented tetrahedra (see ).
A motion of the first kind (orientation-preserving), different from a rotation and a translation, is the product of a translation and a rotation the axis of which is parallel to the direction of the translation (a so-called screwing motion). The theorem was proved by M. Chasles in 1830.
Comments
Any book on linear algebra and analytic geometry can serve as a reference, since both theorems are easy exercises. Another result that also is called Chasles' theorem can be found in [a1]: If the polars of the vertices of a triangle (cf. Polar) do not coincide with the respectively opposite sides, then they meet these sides in three collinear points.
References
| [1] | P.S. Modenov, "Analytic geometry" , Moscow (1969) (In Russian) |
| [a1] | H.S.M. Coxeter, "Projective geometry" , Blaisdell (1964) |
Chasles theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chasles_theorem&oldid=32020