Analytic geometry

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.

Example

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

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Problem: In a convex pentagon ABCDE, the sides have lengths 1, 2, 3, 4, and 5, though not necessarily in

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that order. Let F, G, H, and I be the midpoints of the sides AB, BC, CD, and DE, respectively.

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Let X be the midpoint of segment FH, and Y be the midpoint of segment GI. The length of

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segment XY is an integer. Find all possible values for the length of side AE.

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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).

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

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:Fleft(rac{a}{2},0 ight), Gleft(rac{a+b}{2},rac{e}{2} ight), Hleft(rac{b+c}{2},rac{e+f}{2} ight), Ileft(rac{c+d}{2},rac{f+g}{2} ight), Xleft(rac{a+b+c}{4},rac{e+f}{4} ight), and Yleft(rac{a+b+c+d}{4},rac{e+f+g}{4} ight).

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Using the distance formula,

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

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and

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

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Since XY has to be an integer,

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:AEequiv 0pmod{4}

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(see modular arithmetic) so AE=4.

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