If $a$, $b$ and $c$ are the sides of a triangle with area $\Delta$, prove that $ab + bc + ca \le 4\sqrt3\Delta$ and prove that the equality holds iff the triangle is equilateral.
I tried to approch this problem in terms of the angles of the triangle $\alpha$, $\beta$, $\gamma$. As we know that $\frac{1}{2}ab\sin\gamma = \Delta$. Thus, the above inequality is equivalent to $$\frac{1}{\sin\alpha} + \frac{1}{\sin\beta} + \frac{1}{\sin\gamma} \le 2\sqrt3$$ I tried many ways to proceed further but was unable to do so. Thank you for your help.