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.

(In reply to

Solution by Bractals)

Very nice!

The solution generator you gave (2k+1, k^2+k, k^2+k+1) is linearly independent over second degree polynomials, which proves that all the triangles generated by different values of k will all be non-similar and that there is an infinite number of nontrivial solutions!