Thread: Sine problem from math competition

Suppose that a, b, c are non-zero acute angles such that sin(a − b)/ sin(a + b) + sin(b − c)/ sin(b + c) + sin(c − a)/sin(c + a) = 0. Prove that at least two of a, b, c are equal.


I saw this in a math competition and was perplexed by how to solve the problem. If someone could walk me through it step by step, that would be really helpful.

sin(a+b)= sin(a)cos(b)+cos(a)sin(b)
sin(b+c)= sin(b)cos(c)+cos(b)sin(c)
sin(c+a)=sin(c)cos(a)+cos(c)sin(a)

those are all for the denominators.

I still don't understand. Would you mind explaining how I can prove two of a,b,c equal

Thanks!