Question 171
Let $z_1, z_2, z_3, z_4, z_5, z_6$ be six pairwise different complex numbers whose images $A_1, A_2, A_3, A_4, A_5, A_6$ are consecutive points of the circle with center $O(0,0)$ and radius $r \gt 0$. If $w$ is a solution of the equation $z^2 + z + 1 = 0$ and
$$
z_1 w^2 + z_3 w + z_5 = 0 \quad (I),
$$
$$
z_2 w^2 + z_4 w + z_6 = 0 \quad (II)
$$
prove that: (a) the triangle $A_1A_3A_5$ is equilateral, (b)
$$
|z_1 - z_2| + |z_2 - z_3| + |z_3 - z_4| + |z_4 - z_5| + |z_5 - z_6| + |z_6 - z_1| = 3|z_1 - z_4| = 3|z_2 - z_5| = 3|z_3 - z_6|.
$$
$$
z_1 w^2 + z_3 w + z_5 = 0 \quad (I),
$$
$$
z_2 w^2 + z_4 w + z_6 = 0 \quad (II)
$$
prove that: (a) the triangle $A_1A_3A_5$ is equilateral, (b)
$$
|z_1 - z_2| + |z_2 - z_3| + |z_3 - z_4| + |z_4 - z_5| + |z_5 - z_6| + |z_6 - z_1| = 3|z_1 - z_4| = 3|z_2 - z_5| = 3|z_3 - z_6|.
$$