In triangle $ABC$, angle $\gamma = 120$. Prove that $|\overline{CC'}|=\tfrac{ab}{a+b}$, where $\overline{CC'}$ is symmetral of angle $\gamma$ inside triangle. Look at image.
I can't use areas, becuase we haven't learned them yet. Our teacher says that the trick is to extend $\overline{CC'}$ to have equilateral triangle. But I don't know where to go from that.
i will take your teachers suggestion and extend $CC^\prime$ up to $D$ so that the triangle $BCD$ is an equilateral triangle.
what can you conclude about the lines $BD$ and $AC?$
what about the triangles $ACC^\prime$ and $BDC^\prime?$
i will leave the rest for you to finish off.