Triangle equality

[anemone]

Let $ABC$ be a triangle with $\angle ACB=2\alpha,\, \angle ABC=3\alpha$, $AD$ is an altitude and $AE$ is a median such that $\angle DAE=\alpha$. If $BC=a,\,CA=b$ and $AB=c$, prove that $\dfrac{a}{b}=1+\sqrt{2\left(\dfrac{c}{b}\right)^2-1}$.