betweenness in rays
and are on the same side of line , and
and are on the same side of line .
A point is said to be between and if
there are points and such that is
between and . A point that is between two rays is an
interior point of these rays, but not vice versa in general. A ray
is said to be between rays and
if there is an interior point of and lying
Suppose and is between and . Then
any point on is an interior point of and ;
any point on the line containing that is an interior point of and must be a point on ;
if is defined as above, then any point between and is between and .
There are no rays between two opposite rays.
If is between and , then is not between and .
If has a ray between them, then and must lie on the same half plane of some line.
The converse of the above statement is true too: if are distinct rays that are not opposite of one another, then there exist a ray such that is between and .
- 1 D. Hilbert, Foundations of Geometry, Open Court Publishing Co. (1971)
- 2 K. Borsuk and W. Szmielew, Foundations of Geometry, North-Holland Publishing Co. Amsterdam (1960)
- 3 M. J. Greenberg, Euclidean and Non-Euclidean Geometries, Development and History, W. H. Freeman and Company, San Francisco (1974)
|Title||betweenness in rays|
|Date of creation||2013-03-22 15:33:05|
|Last modified on||2013-03-22 15:33:05|
|Last modified by||CWoo (3771)|
|Defines||between two rays|