Geometry
# Solving Triangles

In triangle $ABC$, let $a, b$ and $c$ be the lengths of the legs of a triangle opposite vertices $A, B, C$ respectively. Let $\alpha, \beta, \gamma$ be the values of the respective angles. What is the value of

$a ( \sin \beta - \sin \gamma) + b ( \sin \gamma - \sin \alpha ) + c ( \sin \alpha - \sin \beta )?$

Triangle $ABC$ satisfies $2\sin B=\sin A+\sin C$. If $a+c=30$, what is the value of $b$?

**Details and assumptions**

$a$, $b$ and $c$ are the lengths of the sides opposite to the vertices $A$, $B$ and $C$, respectively.