Express $AB$, $BC$ and $AC$ over $\sin \alpha$, $\cos \gamma$ and $CC_1$ (note that $AA_1$, $BB_1$ and $CC_1$ are heights, $\alpha =$ angle $BAC$ and $\gamma=$ angle $ACB$). What I tried for now is expressing $\sin \alpha$ and $\cos \gamma$, and i got $\sin \alpha = \frac{CC_1}{AC} = \frac{BB_1}{AB}$ and $\cos \gamma = \frac{A_1C}{AC}=\frac{B_1C}{BC}$, but that leads me nowhere.
We have that $AC=\frac{CC_1}{\sin \alpha}$, we also have that $BC=\frac{CC_1}{\sin \beta}$.
However $\sin \beta=\sin (\pi-(\alpha+\gamma))=\sin (\alpha+\gamma)=\sin \alpha \cos \gamma+\cos \alpha \sin \gamma$. Now we can express $BC$ as required (use identity $\cos^2 \alpha +\sin^2 \alpha=1$ to find $\cos \alpha$, etc.).
Finally use law of sines to express $AB$: $\frac{BC}{\sin \alpha}=\frac{AB}{\sin \gamma}$.