<s>[INST] If $x\sin A=y\sin(A+2π/3) =z\sin(A+4π/3),$ derive a relation among $x, y, z$ by eliminating $A$ 
If $x\sin A=y\sin(A+2π/3) =z\sin(A+4π/3),$ derive a relation among $x, y, z$ by eliminating $A$

This problem was bothering me for a while, and I finally could not solve it. I tried taking the whole equation as $k$ but the calculation was a mess. Would someone please help me to find a solution using a simpler approach? 
 [/INST] Consider (first & second terms) $$x\sin A=y\sin\left(A+\frac{2\pi}{3}\right)$$
$$x\sin A=y\left(\sin A\cos \frac{2\pi}{3}+\cos A\sin\frac{2\pi}{3}\right)$$
$$x=y\left(\frac{\sin A\cos \frac{2\pi}{3}}{\sin A}+\frac{\cos A\sin\frac{2\pi}{3}}{\sin A}\right)$$
$$x=y\left(-\frac12+\cot A \frac{\sqrt3}{2}\right)$$
$$\cot A=\frac{2}{\sqrt3}\left(\frac xy+\frac12\right)\tag 1$$
Similarly, consider (first & third terms) $$x\sin A=z\sin\left(A+\frac{4\pi}{3}\right)$$
$$x\sin A=z\left(\sin A\cos \frac{4\pi}{3}+\cos A\sin\frac{4\pi}{3}\right)$$
$$x=z\left(\frac{\sin A\cos \frac{4\pi}{3}}{\sin A}+\frac{\cos A\sin\frac{4\pi}{3}}{\sin A}\right)$$
$$x=z\left(-\frac12+\cot A \left(-\frac{\sqrt3}{2}\right)\right)$$
$$\cot A=-\frac{2}{\sqrt3}\left(\frac xz+\frac12\right)\tag 2$$
Equating (1) & (2), we get
$$\frac{2}{\sqrt3}\left(\frac xy+\frac12\right)=-\frac{2}{\sqrt3}\left(\frac xz+\frac12\right)$$
$$\frac xy+\frac xz+1=0$$
Hope above is the required relation among $x, y$ & $z$
</s>