<s>[INST] Find $AC$ WITHOUT using Cosine law... 
In the following figure find side $AC$ WITHOUT using Cosine law:
  

There is a simple solution involving Cosine law,but is it possible to calculate $AC$ without using it?? 
 [/INST] Let $DK$ be an altitude of $\Delta ABD$, $AC=x$ and $BD=y$.
Thus, $$DK=\frac{y\sqrt3}{2},$$ 
$$BK=\frac{y}{2}$$ and
$$BC=\sqrt{x^2-1}.$$
Thus, since $$\frac{AB}{BK}=\frac{AC}{CD}$$ and
$$\frac{BC}{DK}=\frac{AC}{AD},$$ we got the following system:
$$\frac{1}{\frac{y}{2}}=\frac{x}{1}$$ and
$$\frac{\sqrt{x^2-1}}{\frac{y\sqrt3}{2}}=\frac{x}{x+1}.$$
Thus, $$\frac{\sqrt{x^2-1}}{\frac{\sqrt3}{x}}=\frac{x}{x+1}$$ or
$$\sqrt{x^2-1}(x+1)=\sqrt3$$ or
$$x^4+2x^3-2x-4=0$$ or
$$(x+2)(x^3-2)=0$$ or
$$x=\sqrt[3]{2}.$$
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