A system of plane geometry where the position of a point is described by A point at (0,0) is positioned at the intersection of the x and y axes.
Compare with polar geometry.
Coordinate geometry is a way of attacking geometrical problems, attempting to reduce them to mere algebra. This is done by describing all the points in the space with a set of coordinates. The idea was developed in the 17th century and is attributed to Rene Descartes.

To begin with you choose a system of coordinates. Most of the time old-fashioned Cartesian coordinates will do (at least if you are dealing with an Euclidean space), but it may be that a suitable choice (skewed axes, polar coordinates, cylindrical coordinates or whatever) will make the calculations easier.
Then you have to choose your reference: origin, directions of axes etc. Again this is done in a way to make the calculations as simple as possible.

Once the coordinates are chosen we can translate the given problem into algebraic terms. The distance between two points is a function of their coordinates, the angles in a triangle are trigonometric functions of the coordinates of the vertices etc. Every condition on the points and lines and surfaces can be interpreted in terms of the coordinates. Sit down and do the algebra and the answer will hopefully come out in the end.

In Euclidean geometry the alternative available is to use Euclidean methods: theorems of congruence and similarity and all their derivatives. With your average 2D problem (involving a few triangles, parallel lines and circles) this approach is likely to give a more elegant solution than coordinate methods - provided that you can find it :-) The coordinate approach is in this case something of a brute force method, and often it only shows that something is true, but not why.
In general, however, coordinate methods are much more powerful than Euclidean ones, which is what makes Descartes's invention of them so important. By describing curves by functions in terms of their coordinates the full force of calculus can be brought to bear, and problems which are otherwise mindbogglingly difficult are made trivial (go ahead and try finding the volume of the volume of rotation of a parabola without using calculus, like Archimedes did).

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