Length of triangle madness!

Algebra Level pending

Let \(a,b,c\) be the length of a triangle, such that \(a>b>c>0\), \(b\) is an integer and satisfy these equations:

  • \(a+c = 2b\)
  • \(a^{2} + b^{2} + c^{2} = 84\)

If the value of \(a+b-c\) can be written as \(x+y\sqrt{z}\) for integers \(x,y,z\) and \(z\) is square-free, find the value of \(x^{3}+y^{3}+z^{3}\)


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