Rational triangles and elliptic curves

One of the many beauties of elliptic curves is their blend of arithmetic and geometry. Studying elliptic curves can lead to insights into many parts of number theory, including finding rational right triangles with integer areas. For example, the integer 6 is the area of the right triangle with sides 3, 4, and 5; whereas 5 is the area of a right triangle with sides 3/2, 20/3, and 41/6. In fact, we can translate the problem of finding rational right triangles with a given area into a question about rational solutions of specific elliptic curves. In addition, the arithmetic of elliptic curves provides us with information about the existence of triangles with rational side lengths that share either: a common side length, the same area, or the same perimeter. Students are expected to study basic properties of elliptic curves and use them to investigate the existence of certain types of rational triangles. 
 
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