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René Descartes is popularly regarded as having introduced the foundation for the methods of analytic geometry in 1637 in the appendix titled ''Geometry'' of the titled ''Discourse on the Method of Rightly Conducting the Reason in the Search for Truth in the Sciences'', commonly referred to as '' Discourse On Method ''. This work, written in his native language ( French ), and its philosophical principles, provided the foundation for Calculus in Europe.


IMPORTANT THEMES OF ANALYTICAL GEOMETRY



Many of these problems involve Linear Algebra


EXAMPLE


Here is an example of a problem from the USAMTS that can be solved via analytic geometry:

Problem: In a convex pentagon ABCDE, the sides have lengths 1, 2, 3, 4, and 5, though not necessarily in
that order. Let F, G, H, and I be the midpoints of the sides AB, BC, CD, and DE, respectively.
Let X be the midpoint of segment FH, and Y be the midpoint of segment GI. The length of
segment XY is an integer. Find all possible values for the length of side AE.

Solution: Let A, B, C, D, and E be located at A(0,0), B(a,0), C(b,e), D(c,f), and E(d,g).

Using the Midpoint formula, the points F, G, H, I, X, and Y are located at

:F\left( rac{a}{2},0 ight), G\left( rac{a+b}{2}, rac{e}{2} ight), H\left( rac{b+c}{2}, rac{e+f}{2} ight), I\left( rac{c+d}{2}, rac{f+g}{2} ight), X\left( rac{a+b+c}{4}, rac{e+f}{4} ight), and Y\left( rac{a+b+c+d}{4}, rac{e+f+g}{4} ight).

Using the Distance formula,

:AE=\sqrt{d^2+g^2}

and

:XY=\sqrt{ rac{d^2}{16}+ rac{g^2}{16}}= rac{\sqrt{d^2+g^2}}{4}.

Since XY has to be an Integer ,
:AE\equiv 0\pmod{4}
(see Modular Arithmetic ) so AE=4.


OTHER USES


Analytic geometry, for Algebraic Geometers , is also the name for the theory of (real or) Complex Manifold s and the more general '''analytic spaces''' defined locally by the vanishing of Analytic Function s of Several Complex Variables (or sometimes real ones). It is closely linked to algebraic geometry, especially through the work of Jean-Pierre Serre in '' GAGA ''. It is strictly a larger area than algebraic geometry, but studied by similar methods.