$\triangle ABC$ is isosceles iff $a \cos B+b \cos C+c \cos A = \frac12(a+b+c)$

Question

Prove that $\triangle ABC$ is isosceles if and only if $$a \cos B+b \cos C+c \cos A = \frac12(a+b+c)$$

Answer 1

Hint: Use that the left-hand side of your equation is given by $$a\left(\frac{c^2+a^2-b^2}{2ac}\right)+b\left(\frac{a^2+b^2-c^2}{2ab}\right)+c\left(\frac{b^2+c^2-a^2}{2bc}\right)=$$