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Example:3D Geometry- Prove Points are Collinear

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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:








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".



In neutral geometry, the distance between (x1) and (x2) 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 (x1, y1) and (x2, y2) is given by:

Similarly, given points (x1, y1, z1) and (x2, y2, z2) 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.




The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.