Consider a triangle with sides of length 5, 6, 7. If you square the area of that triangle, you get 216, a perfect cube.

Are there other triangles (not geometrically similar to the first triangle) with integral sides whose area squared is a perfect cube? Find one such triangle, or prove no others exist.

Let the sides of the triangle be

    a = 2k+1
    b = k^2+k          k>0
    c = k^2+k+1     

then the semiperimeter is

    s = (a+b+c)/2 = (k+1)^2

then the area squared is

    (Area)^2 = s(s-a)(s-b)(s-c)

             = [(k+1)^2][k^2][k+1][k]

             = [k(k+1)]^3  

This does not list all the triangles, but enough.

Edited on March 22, 2007, 6:57 pm

Posted by Bractals on 2007-03-21 21:17:25