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(sa)(sb)(sc)
= [(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 20070321 21:17:25 