<s>[INST] $\sin{\frac{A+B}{2}}+\sin{\frac{B+C}{2}}+\sin{\frac{C+A}{2}} > \sin{A}+\sin{B}+\sin{C}. $ Help me please to prove that: 
for any $\triangle ABC$ we have the following inequality: $$\sin{\frac{A+B}{2}}+\sin{\frac{B+C}{2}}+\sin{\frac{C+A}{2}} > \sin{A}+\sin{B}+\sin{C}. $$
It's about convexity ? 
thanks :)
 [/INST] $\sin A+\sin B=\sin(\frac{A+B}{2}+\frac{A-B}{2})+\sin(\frac{A+B}{2}-\frac{A-B}{2})=2\sin\frac{A+B}{2}\cos\frac{A-B}{2}\leqslant2\sin\frac{A+B}{2}$. Similarly, $\sin B+\sin C\leqslant2\sin\frac{B+C}{2},\sin C+\sin A\leqslant2\sin\frac{A+C}{2}$. Add them together yields the result.
Note that the inequality holds is due to $\frac{A+B}{2}\in(0,\frac{\pi}{2})$ and similar for the rest.
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