Question:

Step 1: Transform the given equation and analyze the modulus. The given equation is $z^6 = (z+1)^6$. First, we note that $z \neq 0$ (otherwise $0^6 = (0+1)^6 \implies 0=1$, false) and $z+1 \neq 0$ (otherwise $(-1)^6 = (-1+1)^6 \implies 1=0$, false). We can rewrite the equation as: $$ \left(\frac{z}{z+1}\right)^6 = 1 $$ Taking the modulus of both sides, we get: $$ \left|\left(\frac{z}{z+1}\right)^6\right| = |1| $$ $$ \left|\frac{z}{z+1}\right|^6 = 1 $$ Since the modulus is a non-negative real number, we have: $$ \left|\frac{z}{z+1}\right| = 1 $$ This implies: $$ |z| = |z+1| $$ Step 2: Determine the locus of the roots. The condition $|z| = |z+1|$ means that any complex number $z$ satisfying this condition is equidistant from the origin $(0)$ and the point $(-1)$ in the complex plane. Geometrically, the locus of such points is the perpendicular bisector of the line segment joining $0$ and $-1$. The midpoint of $0$ and $-1$ is $\frac{0 + (-1)}{2} = -\frac{1}{2}$. The perpendicular bisector of the segment joining $0$ and $-1$ is the vertical line $\operatorname{Re}(z) = -\frac{1}{2}$. Therefore, all roots of the given equation lie on the line $x = -\frac{1}{2}$ in the complex plane. Step 3: Analyze the given expression using the locus of the roots. Let $z_1, z_2, z_3$ be three distinct roots of the equation. From Step 2, we know that all these roots lie on the vertical line $\operatorname{Re}(z) = -\frac{1}{2}$. Let $z_k = -\frac{1}{2} + iy_k$ for $k=1,2,3$, where $y_1, y_2, y_3$ are distinct real numbers. We need to find $\arg\left(\frac{z_1 - z_3}{z_2 - z_3}\right)$. Let's compute the difference between the roots: $z_1 - z_3 = \left(-\frac{1}{2} + iy_1\right) - \left(-\frac{1}{2} + iy_3\right) = iy_1 - iy_3 = i(y_1 - y_3)$ $z_2 - z_3 = \left(-\frac{1}{2} + iy_2\right) - \left(-\frac{1}{2} + iy_3\right) = iy_2 - iy_3 = i(y_2 - y_3)$ Now, form the ratio: $$ \frac{z_1 - z_3}{z_2 - z_3} = \frac{i(y_1 - y_3)}{i(y_2 - y_3)} = \frac{y_1 - y_3}{y_2 - y_3} $$ Since $y_1, y_2, y_3$ are real numbers, the ratio $\frac{y_1 - y_3}{y_2 - y_3}$ is a real number. Also, since $z_1, z_2, z_3$ are distinct roots, $y_1 \neq y_3$ and $y_2 \neq y_3$, so the numerator and denominator are non-zero. Thus, the expression $\frac{z_1 - z_3}{z_2 - z_3}$ is a non-zero real number. Step 4: Determine the possible values of the argument. For any non-zero real number $X$: * If $X > 0$, then $\arg(X) = 0$. * If $X < 0$, then $\arg(X) = \pi$. Since the ratio $\frac{y_1 - y_3}{y_2 - y_3}$ can be either positive or negative depending on the relative order of $y_1, y_2, y_3$, the argument of this ratio can be $0$ or $\pi$. The final answer is $\boxed{0, \pi}$.