relationship between complex numbers

Question

Consider the following:

1. Two equilateral triangles inscribed in a circle.
2. The vertices of the large triangle are the geometric images of the three cubic roots of $z$ (a complex number).
3. The small triangle vertices are the midpoints of the sides of the larger triangle.
4. The vertices of the small triangle are the geometric images of the three cubic roots of $w$ (another complex number).

What is the relationship between $w$ and $z$?

Answer 1

Let $a^3=z$, $b^3=w$, and $r^3=1$, where $r = \exp\frac{2i\pi}{3}$ is the primitive cubic root of unity.

Then the vertices of the outer triangle are $a, ar, ar^2$, those of the inner triangle are $b, br, br^2$, and, wlog, $b=\frac{a+ar}{2}$.

Now, $$w=b^3=a^3(\frac{1+r}{2})^3=z\frac{1+3r+3r^2+r^3}{8}=-\frac{z}{8}$$

because $r^3=1$ and $1+r+r^2=\frac{1-r^3}{1-r}=0$.