An Algebraic Geometry Problem!

Algebra Level 4

In a \(\Delta ABC\), let \(a,\ b,\ c\) be the lengths of the sides of the triangle such that:

  • \(b^2=\dfrac{2a^4}{a^4+1}\)
  • \(c^2=\dfrac{3b^6}{b^8+b^4+1}\)
  • \(a^2=\dfrac{4c^8}{c^{12}+c^8+c^4+1}\)

Find the value of \(\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}\) ?

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