Given the three numbers z1 z2 z3 show that these complex numbers are vertices of an equilateral triangle inscribed in a circle?

Answer 1

An equilateral triangle is always inscribed in a circle.

This means that if you can prove that z1, z2 and z3 are the vertices of an equilateral triangle, they automatically lie on a circle subscribing it.

Compute |z1-z2|, |z1-z3| and |z2-z3|. These need to be equal for z1, z2 and z3 to lie on an equilateral triangle. If not, they aren't lying on an equilateral triangle.

for z=a+ib, |z| = (a^2+b^2)^(1/2).

To find the center c of the circle, note that (z1-c)+(z2-c)+(z3-c) = 0, hence,

c = (z1+z2+z3)/3.