KöMaL 2023 B5322

Prove that if $\frac{\cos\alpha}{s-b}-\frac{\cos\beta}{s-a}=\frac{\cos\alpha-\cos\beta}{s-c}$ in a triangle, with conventional notations, then the triangle is right-angled or isosceles. (As usual, $s$ denotes the semi-perimeter of the triangle.)