Abstract
This chapter discusses the concepts of analytic geometry, which was developed by René Descartes and in which he combined the techniques of algebra with those of geometry. Analytic geometry enables one to apply algebraic methods and equations to the solution of problems in geometry and, conversely, to obtain the geometric representations of algebraic equations. The chapter presents a formula for the coordinates of the midpoint of a line segment and discusses the use of distance and midpoint formulas as tools to illustrate the usefulness of analytic geometry by proving a number of general theorems from plane geometry. The power of the methods of analytic geometry is also very well demonstrated in a study of the conic sections. It is found in the course of that study that (a) a geometric definition can be converted into an algebraic equation and (b) an algebraic equation can be classified by the type of graph it represents.
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