Question 19
(USS 2) Let $\alpha,\beta,\gamma$ be positive real numbers such that $\alpha+\beta+\gamma\lt \pi$, $\alpha+\beta\gt \gamma$, $\beta+\gamma\gt \alpha$, $\gamma+\alpha\gt \beta$. Prove that with the segments of lengths $\sin\alpha,\sin\beta,\sin\gamma$ we can construct a triangle and that its area is not greater than
$$\frac18(\sin2\alpha+\sin2\beta+\sin2\gamma).$$
$$\frac18(\sin2\alpha+\sin2\beta+\sin2\gamma).$$