Consider pairwise distinct complex numbers z1,z2,z3,z4,z5,z6 whose images A1,A2,A3,A4,A5,A6 respectively are succesive points on the circle centered at O(0,0) and having radius r>0.
If w is a root of the equation z2+z+1=0 and the next equalities hold z1w2+z3w+z5=0z2w2+z4w+z6=0 prove thata) Triangle A1A3A5 is equilateral
b) ∣z1−z2∣+∣z2−z3∣+∣z3−z4∣+∣z4−z5∣+z5−z6∣+∣z6−z1∣=3∣z1−z4∣=3∣z2−z5∣=3∣z3−z6∣.