Enumerating Pythagorean Triangles

There is a one-to-one correspondence between positive rational numbers less than 1 and points with positive rational coordinates on the unit circle. This correspondence is achieved by joining the point with and extending the line to intersect the unit circle at as shown in this Demonstration. As any integral solution of the equation corresponding to a Pythagorean triangle can be put in the form , we can associate Pythagorean triangles with points with positive rational coordinates on the unit circle. This Demonstration shows the rational number and its associated Pythagorean triangle. By varying , can you find the only Pythagorean triangle with a side equal to 2009 that exists in the given range? Alas, the first rational with a part equal to 2009 is 30/2009 and it occurs at , too far out of our range .

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