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Appropriate Area (Posted on 2007-03-21) Difficulty: 3 of 5
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.

No Solution Yet Submitted by Gamer    
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Solution Solution | Comment 3 of 4 |

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

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