Chasles theorem

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[edit]

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[edit]

[1] |  P.S. Modenov, "Analytic geometry" , Moscow (1969) (In Russian)

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[a1] |  H.S.M. Coxeter, "Projective geometry" , Blaisdell (1964)