# Article

# Article

- Mathematics
- Geometry
- Analytic geometry

- Mathematics
- Algebra and number theory
- Analytic geometry

# Analytic geometry

Article By:

**Blumenthal, Leonard M. **Formerly, Department of Mathematics, University of Missouri, Columbia, Missouri.

Last reviewed:2014

DOI:https://doi.org/10.1036/1097-8542.032000

- The plane
- Loci and equations
- Equations of lines
- Angle between two lines

- Area of a triangle
- Linear combinations
- Circle
- Conic sections

- Three-dimensional space
- Special surfaces
- n-Dimensions
- Additional Readings

**A branch of mathematics in which algebra is applied to the study of geometry.** Because algebraic methods were first systematically applied to geometry in 1637 by the French philosopher-mathematician René Descartes, the subject is also called cartesian geometry. The basis for an algebraic treatment of geometry is provided by the existence of a one-to-one correspondence between the elements, “points” of a directed line *g*, and the elements, “numbers,” that form the set of all real numbers. Such a correspondence establishes a coordinate system on *g*, and the number corresponding to a point of *g* is called its coordinate. The point *O* of *g* with coordinate zero is the origin of the coordinate system. A coordinate system on *g* is cartesian provided that for each point *P* of *g*, its coordinate is the directed distance $\stackrel{\u2015}{\mathrm{OP}}$. Then all points of *g* on one side of *O* have positive coordinates (forming the positive half of *g*) and all points on the other side have negative coordinates. The point with coordinate 1 is called the unit point. Since the relation $\stackrel{\u2015}{\mathrm{OP}}$ + $\stackrel{\u2015}{\mathrm{PQ}}$ = $\stackrel{\u2015}{\mathrm{OQ}}$ is clearly valid for each two points *P*, *Q*, of directed line *g*, then $\stackrel{\u2015}{\mathrm{PQ}}$ = $\stackrel{\u2015}{\mathrm{OQ}}$ − $\stackrel{\u2015}{\mathrm{OP}}$ = *q* − *p*, where *p* and *q* are the coordinates of *P* and *Q*, respectively. Those points of *g* between *P* and *Q*, together with *P*, *Q*, form a line segment. In analytic geometry it is convenient to direct segments, writing *PQ* or *QP* accordingly as the segment is directed from *P* to *Q* or from *Q* to *P*, respectively. To find the coordinate of the point *P* that divides the segment *P*_{1}*P*_{2} in a given ratio *r*, put $\frac{\stackrel{\u2015}{P{}_{\text{1}}P}}{\stackrel{\u2015}{P{}_{\text{2}}P}}$ = *r*. Then $\frac{\text{(}x\text{\u2212}x{}_{\text{1}}\text{)}}{\text{(}x\text{\u2212}x{}_{\text{2}}\text{)}}$ = *r*, where *x*_{1}, *x*_{2}, *x* are the coordinates of *P*_{1}, *P*_{2}, *P*, respectively, and solving for *x* gives *x* = $\frac{\text{(}x{}_{\text{1}}\text{\u2212}\mathrm{rx}{}_{\text{2}}\text{)}}{\text{(1 \u2212}r\text{)}}$. Clearly *r* is negative for each point between *P*_{1}, *P*_{2} and is positive for each point of *g* external to the segment. The midpoint of the segment divides it in the ratio −1, and hence its coordinate *x* = $\frac{\text{(}x{}_{\text{1}}\text{+}x{}_{\text{2}}\text{)}}{\text{2}}$.* See also: ***Mathematics**

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