Explanation:
Collinear
points
Three points
are said to be collinear if they lie on the same line. In the geometry of space, this is the degenerate
condition where three points do not determine a plane. The concept of collinearity is thus usually derived from a
presumption of lines being in the geometry. However, in synthetic geometry it has been known since 1900
that collinearity can be made a defined concept, and the notion of a line can
be based upon it as a set of collinear points:
Consider the reflection which swaps the points in a
plane that are equal perpendicular distances from a given line L. The fixed points of the reflection are the
points of L. Using d(u,v) to denote the distance between points u and v and selecting a pair of points a and b on L, note that if the reflection
swaps x with y, then
d(x, a) = d(y,a) and d(x,b)
= d(y,b).
The fixed point property of L can
be expressed by saying that x and y are the
same point:
http://en.wikipedia.org/wiki/Line_(geometry)#Collinear_points
Distance
Distance is a numerical
description of how far apart objects are. In physics or everyday discussion, distance may
refer to a physical length, or estimation based on other criteria (e.g.
"two counties over"). In mathematics,
a distance function or metric is a generalization of the concept of
physical distance. A metric is a function that behaves according to a specific
set of rules, and provides a concrete way of describing what it means for
elements of some space to be "close to" or "far away from"
each other.
In most cases, "distance from A to B" is interchangeable
with "distance between B and A".
Geometry
In neutral geometry, the
distance between (x_{1}) and (x_{2}) is the
length of the line segment between them:
In analytic geometry, the
distance between two points of the xy-plane can be found using the
distance formula. The distance between (x_{1}, y_{1}) and (x_{2}, y_{2}) is given by:
Similarly, given points (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) in three-space, the
distance between them is:
(Our solved example in mathguru.com uses
this concept)
These formulae
are easily derived by constructing a right triangle with a leg on the hypotenuse of another (with the other leg orthogonal to the plane that contains the 1st triangle)
and applying the Pythagorean Theorem.
http://en.wikipedia.org/wiki/Distance
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