Analytic geometry

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Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra. Usually the Cartesian coordinate system is applied to manipulate equations for planes, lines, curves, and circles, often in two and sometimes in three dimensions of measurement. As taught in school books, analytic geometry can be explained more simply: it is concerned with defining geometrical shapes in a numerical way, and extracting numerical information from that representation. The numerical output, however, might also be a vector or a shape. Some consider that the introduction of analytic geometry was the beginning of modern mathematics.

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.

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Important themes of analytical geometry

vector space
definition of the plane
distance problems
the dot product, to get the angle of two vectors
the cross product, to get a perpendicular vector of two known vectors (and also their spatial volume)
intersection problems

Many of these problems involve linear algebra

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Example

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

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

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

Using the midpoint formula, the points <math>F</math>, <math>G</math>, <math>H</math>, <math>I</math>, <math>X</math>, and <math>Y</math> are located at

<math>F\left(\frac{a}{2},0\right)</math>, <math>G\left(\frac{a+b}{2},\frac{e}{2}\right)</math>, <math>H\left(\frac{b+c}{2},\frac{e+f}{2}\right)</math>, <math>I\left(\frac{c+d}{2},\frac{f+g}{2}\right)</math>, <math>X\left(\frac{a+b+c}{4},\frac{e+f}{4}\right)</math>, and <math>Y\left(\frac{a+b+c+d}{4},\frac{e+f+g}{4}\right).</math>

Using the distance formula,

and

Since <math>XY</math> has to be an integer,

<math>AE\equiv 0\pmod{4}</math>

(see modular arithmetic) so <math>AE=4</math>.

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