Points, Lines, And Planes
Points, lines, and planes are primitive terms; no attempt is made to define them. They do have properties, however, which can be explicitly described. Among the most important of these properties are the following:
Two distinct points determine exactly one line. That line is the shortest path between the two points. Bricklayers use these properties when they stretch a string from corner to corner to guide them in laying bricks.
Two points also determine a ray, a segment, and a distance, symbolized for points A and B by AB (or BA when B is the endpoint), AB, and AB respectively. (Some authors use AB to symbolize all of these, leaving it to the reader to know which is meant.) Three non-collinear points determine one and only one plane.
The photographer's tripod exploits this to hold the camera steady; the chair on an uneven floor rocks back and forth between two different planes determined by two different combinations of the four legs.
If two points of a line lie in a plane, the entire line lies in the plane. It is this property which makes the plane "flat." Two distinct lines intersect in at most one point; two distinct planes intersect in at most one line. If two coplanar lines do not intersect, they are parallel. Two lines which are not coplanar cannot intersect and are called "skew" lines. Two planes which do not intersect are parallel.
A line which does not lie in a plane either intersects that plane in a single point, or is parallel to the plane.