A triangle has sides $a,b,c$ and angles $\alpha,\beta,\gamma$ such that: $ a \,\cos\beta + b \, \cos\gamma+ c \, \cos\alpha = \frac{a+b+c}{2}$ Prove that the triangle is isosceles.
I tried writing $\cos$ in terms of the sides (using the cosine theorem), for example $ \cos\alpha= \frac{a^2-b^2-c^2}{2ab}$. I get the following equality: $a^3(b-c)+b^3(c-a)+c^3(a-b)=0$
Maybe if I use the triangle inequality in a smart way, I can prove it, but I don't know.