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Difference between revisions of "Chasles theorem"

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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021810/c0218101.png" /> are three arbitrary points on a straight line, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021810/c0218102.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021810/c0218103.png" /> 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 ).
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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 ).
  
 
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.
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Modenov,  "Analytic geometry" , Moscow  (1969)  (In Russian)</TD></TR></table>
 
 
 
  
 
====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. [[Polar|Polar]]) do not coincide with the respectively opposite sides, then they meet these sides in three collinear points.
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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>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Modenov,  "Analytic geometry" , Moscow  (1969)  (In Russian)</TD></TR>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,  "Projective geometry" , Blaisdell  (1964)</TD></TR>
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</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)
How to Cite This Entry:
Chasles theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chasles_theorem&oldid=14846
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article